Google has rolled out a major update for Flow Music Spaces, adding AI-powered song editing, stem splitting, image generation, ...

Joan Birman popularized the Burau representation problem decades ago. Now she and two colleagues have a proof that reveals ...

Learn how to determine the domain and range of a function given the graph of the function. Since the domain of a function is the set of all x-values, we will want to identify how far left the graph ...

👉 Learn how to determine the end behavior of the graph of a polynomial function. To do this we will first need to make sure we have the polynomial in standard form with descending powers. We will ...

The original version of this story appeared in Quanta Magazine. In 1939, upon arriving late to his statistics course at UC Berkeley, George Dantzig—a first-year graduate student—copied two problems ...

Years ago, an audacious Fields medalist outlined a sweeping program that, he claimed, could be used to resolve a major problem in algebraic geometry. Other mathematicians had their doubts. Now he says ...

\[f(x) = x^{-2}\text{ or }f(x) = \frac{1}{x^{2}}\] Both functions have asymptotes at \(x = 0\) and \(y = 0\) for their basic forms. The vertical asymptote at \(x = 0 ...

This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2024. Many real-world networks change dynamically but can be notoriously ...

Nodes of a network have opinions represented by real numbers. Occasionally, two neighbors interact and replace their opinions by their average. This model has applications in engineering and social ...

Graph polynomials serve as powerful invariants that encode both combinatorial and topological features of graphs drawn on surfaces. Beginning with the classical Tutte polynomial for planar graphs, ...

Graphs are used as a model of complex relationships among data in biological science since the advent of systems biology in the early 2000. In particular, graph data analysis and graph data mining ...