In any normal linear algebra class you learn how to solve a system of linear equations using Gaussian elimination methods. But how would solve a system like the following:
$$x^2+y+z=1$$
$$x+y^2+z=1$$
$$x+y+z^2=1$$
In this talk I will go over how one can use Gr\"obner theory to solve a system of nonlinear equations and what assumptions might need to be made to make this possible.
In this work we present and analyse a fully-mixed formulation for the nonlinear model given by the coupling of the Navier-Stokes and Darcy-Forchheimer equations with the Beavers-Joseph-Saffman condition on the interface. Our approach yields non-Hilbertian normed spaces and a twofold saddle point structure for the corresponding operator equation. Furthermore, since the convective term in the Navier-Stokes equation forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms.
Fluid models were developed as an alternative to the Navier-Stokes equations to avoid computational complexity especially in case of turbulent flows. Model errors due to the variation of a model parameter become an immediate concern in different aspects. In this presentation, we discuss two such aspects as the conservation of energy and enstrophy, and the reliability of the model given a parameter value for the so called Time Relaxation Model.
Using an oscillating ratchet geometry, an investigation of the
net fluid transport through the ratchet is undertaken in both Newtonian
and viscoelastic fluids. A detailed description of the numerical model
that couples the moving immersed ratchet structure to the bulk fluid
will be discussed. Numerical results presented will detail how
viscoelasticity enhances the net fluid transport in the ratchet.
Are you interested in becoming a math major or what happens after graduation for math majors? Come and ask our undergraduate seniors!
If you give one of your friends a puzzle to solve, they will often dive right into a trial-and-error approach to finding a solution. For instance, if we have a line of lightbulbs, each of which with a switch that affects its state along with the state of its immediate neighbor bulbs, we can ask if it's possible to turn off all of the bulbs from some starting configuration of on/off. The trial-and-error approach may work for four bulbs, but what if there were 20 bulbs? Luckily for us, some simple linear algebra tools can give us clarity for all possible cases.
The derivative of $f(x,y)$ in the direction $\vec{v}=(v_1,v_2)$, at point $p$, is:
$$(f_x(p),f_y(p)) \cdot (v_1,v_2) = v_1 f_x(p) + v_2 f_y(p) $$
But wait: $f_x(p)$ and $f_y(p)$ measure how $f$ changes in directions of $x$ and $y$. Then the formula above claims that knowing only this piece of information, we can know how $f$ changes in any other direction as well?! How could this be true? I will answer this, and end with a eulogy of derivatives in general!