mathematics

Let $H$ be an infinite-dimensional Hilbert space and let $(x_n)\subset H$ satisfy $$ x_n\rightharpoonup 0 $$ (weakly), and suppose additionally that $$ \|x_n\|\to L $$ for some $L>0$. Let me define ...

Edward O. Thorp is one of those people who really should be better known. His Wikipedia entry describes him as ‘an American mathematics professor, author, hedge fund manager, and blackjack researcher’ before elaborating that he ‘… pioneered the modern applications of probability theory, including the harnessing of very small correlations for reliable financial gain.’ In fact Thorp’s life has been…

There are many ways to formulate the group axioms. A standard definition is this: A group is pair $(G,\circ)$ consisting of a set $G$ and a binary operation $\circ$ on $G$ satisfying the following ...

Ramanujan's 1913 letter looked implausible because it compressed deep mathematics into unsupported claims. Hardy's response mattered because suspicion led to examination rather than dismissal.

A cryptography library had a bottleneck no test could see : encrypting half a megabyte took fifty seconds. Every test passed. They had been passing for months. The cause is a trap that keeps recurring: a correct, well-documented optimisation that fixes the constant and not the order — and whose comment, precisely because it is well written, convinces the reader the problem is already solved. What…

The previous post discussed the motivation for and application of the rank-trace theorem. This post will give a proof. Suppose A is a real symmetric matrix. The rank-trace inequality says where tr is the trace operator, the sum of the elements along the diagonal of the matrix. Terse proof Here’s the proof in a nutshell: diagonalize A […] The post Proof of the rank-trace theorem first appeared on …

A team of mathematicians discovered that five 60-sided dice, arranged with exactly the right numbers, can determine a perfectly fair turn order for any group of players, with zero chance of a tie.

University of Notre Dame mathematics graduate students Tan Özalp and Fuxiang Yang have been awarded Simons Dissertation Fellowships in Mathematics by the Simons Foundation. Created to honor the legacy of foundation co-founder Jim Simons, these fellowships support doctoral students during the final…

Peter Cholak, Professor in the Department of Mathematics, has been awarded the Best Paper Award at the 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Sponsored by the European Association for Theoretical Computer Science (EATCS), the award recognizes the…

You have two nails in the wall and a painting with a string on its back, which can easily be rested on the nails to hang the painting. If you remove one nail, the painting will still hang on the other. But mathematicians said, “We can make it worse.” In 1997 A. Spivak posed the following riddle: Is there a way to hang the painting so that removing either nail will cause the painting to fall? Sinc…

_International Journal of Education, Technology and Science_ 6 (3): 337–358. 2026Mathematical reasoning is a fundamental competency in contemporary mathematics education, yet many junior high school students continue to experience difficulties in constructing logical arguments, connecting mathematical concepts, and solving complex problems. This study examined changes in junior high school studen…