Tagged Questions

Numerical algorithms for problems in analysis and algebra, scientific computation

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Numerical integration over a cube with non-product weight

Numerical integration over an interval with (well-behaved) weight functions is a research area that has received considerable attention in the past centuries. Any cubature formula over a interval ...
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The classical two phase Stefan problems

What is the most commonly used treatment method of the moving interface in the classical two phase Stefan problems with the finite element method. Here I mean the water-ice two phase problem under ...
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Is there a brute force method for determining irreducible representations?

Suppose I have some groups $G_1$, $G_2$, $G_3$, etc... Then the direct product is given by $G = G_1 \times G_2 \times G_3 \ldots$ I know that the sub-representations of a reducible representation ...
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On the computation of generalized eigenvalues of a low-rank approximation using SVD

I have trouble deriving an expression of the generalized eigenvalues of a matrix pair, found in http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4618700 . The setup is the following and can ...
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Solving Schroedinger Equation for the electronic energies of the Molecular Ion Hydrogen H2+ in the Elliptic coordinate system [closed]

Electronic Energies of Molecular Ion Hydrogen $H_2^{+}$ $r_1$ is the distance between the proton $1$ and the electron. $r_2$ is the distance between the proton $2$ and the electron. $R$ is the ...
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Compensated compactness for system of conservation laws?

As far as I knew, the method of compensated compactness can be used only for one-dimensional scalar and $2\times 2$ systems of conservation laws, i.e. $u_t+f(u)_x=0$. But if I understood correctly ...
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The condition number of a scaled Vandermonde matrix

Let $V(x_1,..,x_n)$ be the Vandermonde matrix induced by $x_1,..,x_n$, and let $\tilde{V} := V(\frac{x_1}{h},...,\frac{x_n}{h})$. My intuition says that the condition number should be invariant under ...
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Fit a system of linear ODEs from several experiments

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n=6$ (in any event a number below 10) , and the number of initial vectors $r$ is in the ...
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Level Set Advection with Extension Velocity

We're studying the following system of PDEs for a scalar function $F(x, t)$ with $x \in \mathbb{R}^3$ and $t \in \mathbb{R}$. The function $F(\cdot, t)$ is a level set function for a time-dependent ...
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Solving a nonlinear PDE numerically

I want to solve numerically the following PDE: $$u_x + u_t - (u_{xt})^2 = u(x,t)$$ The boundary conditions are no concern of mine, pick the ones that work. So which numerical method will work for ...
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Solving over-determined system of polynomials

I am trying to solve the following over determined system of polynomials \begin{align} & p_1(x_1,x_2,\ldots,x_n)=0, \\ & p_2(x_1,x_2,\ldots,x_n)=0, \\ & \vdots \\ & p_m(x_1,x_2, \...
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What am I missing in this highly oscillatory integral? [closed]

I want to numerically integrate this equation (in python): $\int_{0}^{\infty}{\rm d}k f(k) J_v(r k)J_v(s k)$, where f(k) is a non-smooth function, and $J_v$ are the Bessel function of the fist kind....
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Are piecewise linear functions dense in $W^{1,\infty}$?

Are piecewise linear functions dense in $W^{1,\infty}$ ?
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Time-stepping numerical scheme for the advection dispersion equation

I am facing a simple (at first glance) problem. I need to implement a numerical scheme for the solution of the first order wave propagation equation with chromatic dispersion included. My original ...
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What is the rate of convergence? [closed]

How quickly does the series defined by $$x_0 = 0, \ x_{n+1} = \frac{x_n^2+1}{2}$$ converge to $1$?
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Finding t vlaue in Bezier curve [closed]

According to this question, I'm looking for some method to find the t value in Quadratic bezier curve equation: $$B(t)=P_0+t(1-t)P_1+t^2P_2 \space \space where \space 0 ≤ t ≤ 1$$ In this ...
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Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$-a\Delta u + f(u) = 0,$$ $$u|_\Gamma = u_0$$ by Newton’s method when its convergence is global and monotonic. ...
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Why is there a $\sqrt{5}$ in Hurwitz's Theorem?

Hurwitz's theorem is an extension of Minkowski's Theorem and deals with rational approximations to irrational numbers. The theorem states: For every irrational number $\alpha$, there are infinitely ...
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Open volumetric time series data set

Does anyone know where I can find a good open volumetric time series data set? I had a look at some of Stanford's open data sets (https://graphics.stanford.edu/data/voldata/ ) But these do not seem ...
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Fixed point iteration on symmetric biconvex function

Suppose $X\subseteq\mathbb{R}^n$ is a convex set and that a function $g(x,y):X\times X\rightarrow\mathbb{R}_+$ is smooth, "strictly biconvex" (strictly convex in $x$ and $y$ independently but not ...
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Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay the same

I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, i.e.,...
I am interested in finding the gap between the two least eigenvalues of $A(t)$, a Hermitian $N\times N$ sparse ‎matrix whose diagonal elements are $a_it+b_i\,(1\leq i\leq N)$, and all off-diagonal non-...