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2
votes
2answers
737 views

Aproximation of a Normal Distribution function

I am reviewing and documenting a software application (part of a supply chain system) which implements an approximation of a Normal Distribution function; the original documentation mentions the ...
0
votes
0answers
181 views

Obtaining precise values from good approximation

Problem: We would like to calculate $S=\sum_{i=1}^{k} c_i x_i$, where $k$ is a constant, $x_i$ are some fixed algebraic numbers, $c_i=\frac{p_i}{q_i}$ are rational numbers such that integers $p_i$ ...
10
votes
4answers
542 views

Finite interpolation by a nondecreasing polynomial

Let $x_1 < x_2 < \ldots < x_n$ and $y_1 < y_2 < \ldots < y_n$ be two sequences of $n$ real numbers. It is well known that there are polynomials that "interpolate" in that ...
6
votes
0answers
422 views

convergence rate in Wiener's approximation theorem

Wiener has the following fantastic results about approximations using translation families: Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in ...
5
votes
1answer
294 views

Sparse approximate representation of a collection of vectors

Suppose I have a collection of $n$ vectors $C \subset \mathbb{F}_2^n$. They are of course spanned by the canonical set of $n$ basis vectors. What I would like to find is a much smaller (~ $\log n$) ...
3
votes
2answers
870 views

Variant of Fermat's last theorem

By Fermat's last theorem, the equation $u^3+v^3=w^3$ has no solutions in positive integers $u,v,w$. Now consider the following variant : call $\rho(x)$ the distance between $x$ and the nearest ...
12
votes
6answers
3k views

Solving NP problems in (usually) Polynomial time?

Just because a problem is NP-complete doesn't mean it can't be usually solved quickly. The best example of this is probably the traveling salesman problem, for which extraordinarily large instances ...
4
votes
1answer
427 views

Interpolation by a function whose second derivative is bounded

I don't know if this is an easy question for specialists in the field. Consider the following interpolation problem : let $\varepsilon >0$, $X$ be a finite set of real numbers and $g$ be a ...
8
votes
3answers
636 views

Approximating with translated Gaussians and low-frequency trig functions

Defining the translated Gaussians by $f_t(x)=\exp(-(x-t)^2)$ for $t,x\in\Bbb{R}$, we showed that the linear span of $\{f_t \mid 0 \le t < \epsilon\}$ is dense in $L^2(\Bbb{R})$, for any ...