Scientific machine learning (SciML) methods such as physics-informed neural networks (PINNs) are used to estimate parameters of interest from governing equations and small quantities of data. However, there has been little work in assessing how well...
This work is concerned with a class of optimal control problems governed by a partial differential equation that are motivated by an application in radiotherapy treatment planning, where the primary design objective is to minimize the volume where a...
Aluminum metal reacts with aqueous iron (II) chloride to form aqueous aluminum chloride and iron metal. What is the stoichiometric coefficient for iron when the chemical equation is balanced using …
Write the balanced equation for the reaction of zinc (II) chloride and lead (II) nitrate, including the correct formulas for each reactant and product. What is the mass of aluminum that can be produced by the …
Solutions of partial differential equations can often be written as surface integrals having a kernel related to a singular fundamental solution. Special methods are needed to evaluate the integral accurately at points on or near the surface. Here we...
Many problems in fluid dynamics are effectively modeled as Stokes flows - slow, viscous flows where the Reynolds number is small. Boundary integral equations are often used to solve these problems, where the fundamental solutions for the fluid veloci...
In this note we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM) generalized biadjoint amplitudes for all $k$ and $n$ on what we call the minimal kinematics. We prove that on the minimal kinematics, the scattering equations on the config...
By means of numerical methods, we solve the equations of motion for the collapse of a shell of baryonic matter, made of galaxies and substructure falling into the central regions of a cluster of galaxies, taking into account the effect of the dynam...
Here we consider the nonlinear evolution of Alfven waves that have been excited by gravitational waves from merging binary pulsars. We derive a wave equation for strongly nonlinear and dispersive Alfven waves. Due to the weak dispersion of the Alfv...
Classical approach of solvability problem has shed much light on what we can solve and what we cannot solve mathematically. Starting with quadratic equation, we know that we can solve it by the quadratic formula which uses square root. Polynomial is...
We introduce a new theoretical framework -- the polarized Houston basis -- to model nonequilibrium dynamics in driven open quantum systems, formulated for use within the quantum master equation. This basis extends conventional Houston states by incor...
Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spati...
Evans developed a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. In an accompanying paper I, we analyzed this theory and summarized it in nine equations. We now...
The algebras of the symmetry operators for the Hamilton-Jacobi and Klein-Gordon-Fock equations are found for a charged test particle moving in an external electromagnetic field in a spacetime manifold, on the isotropic (null) hypersurface of which a...
Versatile mixed finite element methods were originally developed by Chen and Williams for isothermal incompressible flows in "Versatile mixed methods for the incompressible Navier-Stokes equations," Computers & Mathematics with Applications, Volume 8...
We test the consistency with which Simmons' model can predict the local current density obtained for flat metal-vacuum-metal junctions. The image potential energy used in Simmons' original papers had a missing factor of 1/2. Besides this technical is...
In this study, we investigate the numerical stability of the covariant BSSN (cBSSN) formulation proposed by Brown. We calculate the constraint amplification factor (CAF), which is an eigenvalue of the coefficient matrix of the evolution equations of...
Using a three-dimensional hydrodynamic simulation of the collision and an equation of state containing a first order phase transition to the quark-gluon plasma, we study thermal photon production for $Au+Au$ collisions at $E_{lab}=11.5$ AGeV and fo...
The equation of state of the linear sigma model in the mean field approximation is used as input in a relativistic hydrodynamical numerical routine. Longitudinal and transverse energy distributions are calculated and compared with those obtained fr...
It even formats your equation or expression as you enter it, making it easier to ensure you're entering it correctly. It's not quite as robust as Wolfram, but it's very useful for showing the steps f...