We consider the problem of forecasting the next (observable) state of an unknown ergodic dynamical system from a noisy observation of the present state. Our main result shows, for example, that support vector machines (SVMs) using Gaussian RBF kernels can learn the best forecaster from a sequence of noisy observations if (a) the unknown observational noise process is bounded and has a summable ...

We study the statistical mechanics of random surfaces generated by N × N onematrix integrals over anti-commuting variables. These Grassmann-valued matrix models are shown to be equivalent to N ×N unitary versions of generalized Penner matrix models. We explicitly solve for the combinatorics of ’t Hooft diagrams of the matrix integral and develop an orthogonal polynomial formulation of the stati...

We show that if (S,+) is a commutative semigroup which can be embedded in the circle group T, in particular if S = (N,+), then all nonprincipal, strongly summable ultrafilters on S are sparse and can be written as sums in βS only trivially. We develop a simple condition on a strongly summable ultrafilter which guarantees that it is sparse and show that this holds for many ultrafilters on semigr...

Let ∆xf(x, y) = f(x + 1, y) − f(x, y) and ∆yf(x, y) = f(x, y + 1) − f(x, y) be the difference operators with respect to x and y. A rational function f(x, y) is called summable if there exist rational functions g(x, y) and h(x, y) such that f(x, y) = ∆xg(x, y)+ ∆yh(x, y). Recently, Chen and Singer presented a method for deciding whether a rational function is summable. To implement their method ...

Ingliam's (nonregular) summation method (I) is closely connected with prime number theory. An easy limitation theorem for (I) (observed by Hardy) is if Xc, is summable (I) then c " = o(loglog n). We show this result to be best possible. Ingham's summation method (1) [2] (also discovered independently by Wintner [6]) may be defined as follows : A series Ecn will be said to be summable (1) to A i...

Let T : X → X be a real analytic full branch expanding map, where X is a compact connected subset of R with non-empty interior. A well known sufficient condition for the existence of a T -invariant probability measure equivalent to Lebesgue measure is that T has bounded distortion. An alternative sufficient condition is that T has uniformly summable derivatives (see [BJ]). The purpose of this n...

This paper extends some results for the weighted Moore–Penrose inverse A+M,N in Hilbert space to the so-called weighted Minkowski inverse A⊕M,N of an arbitrary rectangular matrix A ∈ Mm,n in Minkowski spaces μ. Four methods are also used for approximating the weighted Minkowski Inverse A⊕M,N . These methods are: Borel summable, Euler–Knopp summable, Newton–Raphson and Tikhonov’s methods. c © 20...