Wolfram|Alpha Examples: Mathematics
Whether it be arithmetic, algebra, calculus, differential equations or anything in between, Wolfram|Alpha is up to the challenge. Get help with math homework, solve specific math problems or find …
Myron Mathisson: what little we know of his life
Myron Mathisson (1897-1940) was a Polish Jew known for his work on the equations of motion of bodies in general relativity and for developing a new method to analyze the properties of fundamental solutions of linear hyperbolic differential equation...
Nonuniform $μ$-dichotomy spectrum and kinematic similarity
For linear nonautonomous differential equations we introduce a new family of spectrums defined with general nonuniform dichotomies: for a given growth rate $μ$ in a large family of growth rates, we consider a notion of spectrum, named nonuniform $μ...
A least-squares meshfree method for the incompressible Navier-Stokes equations: A satisfac…
Incompressible flow solvers based on strong-form meshfree methods represent arbitrary geometries without the need for a global mesh system. However, their local evaluations make it difficult to satisfy incompressibility at the discrete level. Moreove...
Fine-tuning physics-informed neural networks for cavity flows using coordinate transformat…
Physics-informed neural networks (PINNs) have attracted attention as an alternative approach to solve partial differential equations using a deep neural network (DNN). Their simplicity and capability allow them to solve inverse problems for many appl...
Computation of maximal local (un)stable manifold patches by the parameterization method
In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and maximizes the size...
Uniform bounds for solutions to elliptic problems on simply connected planar domains
We consider the singular Liouville equation and the Henon-Lane-Emden problem on simply connected planar domains. We show that any solution to each problem must satisfy a uniform bound on the mass. The same results applies to some systems and more gen...
Desmos | Graphing Calculator
Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.
Derivation of the Lorentz Force Law and the Magnetic Field Concept using an Invariant Form…
It is demonstrated how the right hand sides of the Lorentz Transformation equations may be written, in a Lorentz invariant manner, as 4--vector scalar products. The formalism is shown to provide a short derivation, in which the 4--vector electromag...
Derivation of the Lorentz Force Law, the Magnetic Field Concept and the Faraday-Lenz Law u…
It is demonstrated how the right hand sides of the Lorentz Transformation equations may be written, in a Lorentz invariant manner, as 4--vector scalar products. This implies the existence of invariant length intervals analogous to invariant proper...
Retarded electric and magnetic fields of a moving charge: Feynman's derivation of Liénard-…
Retarded electromagnetic potentials are derived from Maxwell's equations and the Lorenz condition. The difference found between these potentials and the conventional Liénard-Wiechert ones is explained by neglect, for the latter, of the motion-depend...
A Microsoft Word Font for Anti-Matter
One often wishes to quickly add a few overlined characters such as anti-B0 or anti-neutrino to a Microsoft Word document. Underlined characters are straightforward but overlined characters require equation editor which makes small picture files. Th...
Optimal neutron-star mass ranges to constrain the equation of state of nuclear matter with…
Exploiting a very large library of physically plausible equations of state (EOSs) containing more than $10^{7}$ members and yielding more than $10^{9}$ stellar models, we conduct a survey of the impact that a neutron-star radius measurement via elect...
New constraints on radii and tidal deformabilities of neutron stars from GW170817
We explore in a parameterized manner a very large range of physically plausible equations of state (EOSs) for compact stars for matter that is either purely hadronic or that exhibits a phase transition. In particular, we produce two classes of EOSs w...
Impact of magnetic field-driven anisotropies on the equation of state probed in neutron st…
Binary neutron star mergers can produce extreme magnetic fields, some of which can lead to strong magnetar-like remnants. While strong magnetic fields have been shown to affect the dynamics of outflows and angular momentum transport in the remnant, t...
Improving Met Office Weather and Climate Forecasts with Bespoke Multigrid Solvers
At the heart of the Met Office climate and weather forecasting capabilities lies a sophisticated numerical model which solves the equations of large-scale atmospheric flow. Since this model uses semi-implicit time-stepping, it requires the repeated s...
Search for continuous gravitational waves from known pulsars in the first part of the four…
Continuous gravitational waves (CWs) emission from neutron stars carries information about their internal structure and equation of state, and it can provide tests of General Relativity. We present a search for CWs from a set of 45 known pulsars in t...
Controlling Grey Solitons in a Trap
Experimentally observed grey solitons are analytically extracted from a physically viable \GP equation. Associated Lieb and Bogoliubov modes are calculated for these class of solitons. It is observed that, these nonlinear excitations have strong...
Random Shooting of Entangled Particles in Vacuum
The effect of random shooting of particles is considered on the basis of solution of the Schrodinger equation and in terms of the Wigner function. Two-particles description shows, in particular, that initial correlation leads to high velocities of...
akalin/abel-ruffini-topological-proof
This repository contains code for demonstrating Arnold's topological proof of the Abel-Ruffini theorem that there is no general solution in radicals to polynomial equations of degree five or higher. (⭐ 3 | JavaScript)