Invariants, inverses, and composite functions: core tools for analyzing mathematical transformations.
An element a is an invariant of f if f(a) = a. The inverse f⁻¹ satisfies f⁻¹(f(x)) = f(f⁻¹(x)) = x. The composite fg(x) represents f(g(x)) where the image of g is the argument of f.
Essential in physics for steady-state analysis, in cryptography for decryption, and in programming for function pipelines.
Quadric surfaces shape everything from planetary models to architectural marvels.
These diagrams depict the six classic forms - the ellipsoid, elliptical paraboloid, elliptical cone, hyperboloids of one and two sheets, and hyperbolic paraboloid - complete with their cross-sections and defining equations, including x²/a² + y²/b² + z²/c² = 1 for the ellipsoid and y²/b² − x²/a² = z/c (c > 0) for the hyperbolic paraboloid.
Quadric surfaces have practical uses in real life, including hyperboloid cooling towers in power plants, parabolic reflectors in satellite dishes and telescopes, and hyperbolic paraboloid structures in contemporary architecture.
From perceptrons and feed-forward networks to LSTMs, GRUs, GANs, convolutional nets, residual connections, capsule networks, and attention mechanisms, this chart maps major neural network architectures with standardized diagrams and a clear legend for cell types and connections.
These designs power image recognition in medical diagnostics, language translation in virtual assistants, recommendation engines on streaming platforms, and autonomous systems in robotics and self-driving vehicles.
Spelman College’s board of trustees has tapped Ayanna Howard, a dean of engineering at The Ohio State University, to serve as the school’s 12th president.
In an interview with the AJC, Ayanna Howard shares more on her background and plans for Spelman in her new role. Read more: https://t.co/EGGRA1ZQUF
🖋️ : Martha Dalton / AJC
📸 : Natrice Miller for the AJC
Spelman announces that Ayanna Howard will be the new president.
Ayanna Howard is a world-renowned roboticist, AI expert, entrepreneur, and university leader who has spent her career pushing the boundaries of technology while expanding opportunities in STEM education.
🚨BREAKING NEWS: The James Webb Space Telescope confirms that there is something seriously wrong with our understanding of the universe, and reveals that there is an unknown physics.
Platonic and Archimedean polyhedra, the five regular solids and their semi-regular counterparts.
At the top are the five Platonic solids: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Every face is an identical regular polygon and the same number of faces meet at each vertex.
Below are the fifteen Archimedean polyhedra. They keep all vertices identical but allow two or more types of regular polygonal faces, including both left- and right-handed versions of the snub cube and snub dodecahedron.
These shapes turn up throughout the natural and built world, from virus capsids and buckyball molecules to geodesic domes, soccer balls, and efficient structural designs in architecture and materials science.
Vector calculus in its purest form: the gradient, divergence, and curl.
These three operators are the language we use to describe how fields change in space. The gradient shows the direction and rate of steepest increase for a scalar like temperature or pressure. Divergence tells us whether a vector field is spreading out or squeezing in, exactly what you need for fluid flow or electric fields. Curl captures the local rotation or swirling motion inside the field, the reason magnetic fields form loops around currents and why vortices appear in rivers or air.
From weather modeling and aerodynamics to Maxwell’s equations and optimization algorithms, these are the tools that turn raw math into real-world predictions.