# Tagged Questions

95 views

### Stationary Distribution for Markov-like system?

Let $$A= \begin{pmatrix} 0 & a_{1,2} & a_{1,3} \\ a_{2,1} & 0 & a_{2,3} \\ a_{3,1} & a_{3,2} & 0 \end{pmatrix},$$ B= ...
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### Delay Differential Equations Numerical methods

I have a general question about delay differential equations. I know that even simple ones hardly have analytic solutions and mine clearly doesn't have any as it is a system of non-linear delay ...
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### Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)

According to the entry "Differential inequality" of the Encyclopedia of Mathematics http://www.encyclopediaofmath.org/index.php/Differential_inequality the following result is due to Chaplygin ...
256 views

### What are the basis functions for a product space?

Let $X=L^1\left([0,1]^3\right)$, for numerical purpose, what are the possible basis function for $X$? In finite element method, the basis functions are tooth functions, or polynomial functions. Is ...
118 views

### elliptic integral with singularities

I need to calculate elliptic integrals with singularities, up to a huge number of digits (250-1000). The problem is that Wolfram Mathematica can't do so many digits, and Pari intnum doesn't handle ...
266 views

### Rigorous numerics for maxima and minima (one variable)

Let $f:\mathbb{R}_0^+\to \mathbb{R}$ be defined by some combination of the four basic operations and square roots. (The argument of square-roots is assumed is to be non-negative, and the value of ...
126 views

### For what values of the parameter does this function have an elementary anti-derivative?

I am a grad student working on some independent research trying to derive some exact formulas for a particular class of power series. During my study I came across the following integral which would ...
2k views

### The unreasonable effectiveness of Pade approximation

I am trying to get an intuitive feel for why the Pade approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. But what I can't ...
124 views

### approximation methods in integral equations

Recently I was reading about integral equations and I am a beginner in it. There was a constant reference to the non-availability of methods to find the exact solutions and hence lot of approximation ...
841 views

### Easy functions ?

Let $f$ be an analytic function, and suppose that we want to compute $f(x)$. The input consists of the digits of $x$ and the output of a rational number approximating $f(x)$. A function $f$ is called ...
2k views

### Any good books on numerical methods for ordinary differential equations?

I need to find some masters-level exercises about numerical methods for solving ODEs. Are there any good references?
378 views

### Series acceleration for more complicated types of oscillating series

Question: It is well known that the greatest integer function has a Fourier series representation. Since the greatest integer function itself is not periodic, the representation is derived from the ...
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### Minimum time planar paths under a bound on magnitude of acceleration

On a plane, given initial position (x1,y1), initial velocity (u1,v1), final position (x2,y2), and final velocity (u2,v2), compute the solution to x''= cos(z), y''=sin(z) that has these endpoint ...
395 views

### Approximating high-dimensional integrals by low-dimensional ones

This question is motivated by the following naive one: suppose we have a nice subset $X$ of some Euclidean space, say a polyhedron, and a nice $\mathbb{R}$-valued function $f$ on this subset, say a ...
1k views

### Stone-Weierstrass theorem applied to Fourier series

This is a question on Fourier series convergence. The problem is, in the applications of the Stone Weierstrass approximation theorem on wikipedia, there's stated that as a consequence of the theorem ...
934 views

### An inequality on concave functions

Could somebody help me to answer the following question? Let $f:R_+ \rightarrow R_+$ be a nonindentically zero, nondecreasing, continuous, concave function with $f(0)=0$. Do we have that for any ...
508 views

### Runge-Kutta method with c<1

In trying to solve an ODE $y'=f(y,t)$ with a function f that is discontinuous at a subset (codim=1) of $\mathbb R^n$, I am looking for a Runge-Kutta ODE method whose stages do not evaluate $f(x,t)$ at ...
581 views

### best approximation to the LambertW(x) or exp(LambertW(x))

what is the best available approximation ( say up to 10 digits ) for LambertW(x) or exp(LambertW(x)) for x > 2000