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Given the closed interval [0,x] with x>1, let one-dimensional "cars" of unit length be parked randomly on the interval. The mean number M(x) of cars which can fit (without ...

Every irrational number x has an approximation constant c(x) defined by c(x)=lim inf_(q->infty)q|qx-p|, where p=nint(qx) is the nearest integer to qx and lim inf is the ...

Let T_n(x) be an arbitrary trigonometric polynomial T_n(x)=1/2a_0+{sum_(k=1)^n[a_kcos(kx)+b_ksin(kx)]} (1) with real coefficients, let f be a function that is integrable over ...

As Lagrange showed, any irrational number alpha has an infinity of rational approximations p/q which satisfy |alpha-p/q|<1/(sqrt(5)q^2). (1) Furthermore, if there are no ...

The Littlewood conjecture states that for any two real numbers x,y in R, lim inf_(n->infty)n|nx-nint(nx)||ny-nint(ny)|=0 where nint(z) denotes the nearest integer function. ...

Let f(x) be integrable in [-1,1], let (1-x^2)f(x) be of bounded variation in [-1,1], let M^' denote the least upper bound of |f(x)(1-x^2)| in [-1,1], and let V^' denote the ...

The process of approximating a quantity, be it for convenience or, as in the case of numerical computations, of necessity. If rounding is performed on each of a series of ...

Let K subset= C be compact, let f be analytic on a neighborhood of K, and let P subset= C^*\K contain at least one point from each connected component of C^*\K. Then for any ...

There are two distinct entities both known as the Lagrange number. The more common one arises in rational approximation theory (Conway and Guy 1996), while the other refers ...

Let lambda_(m,n) be Chebyshev constants. Schönhage (1973) proved that lim_(n->infty)(lambda_(0,n))^(1/n)=1/3. (1) It was conjectured that the number ...