We consider the Complex Stone-Weierstrass Property (CSWP), which is the complex version of the Stone-Weierstrass Theorem. If X is a compact subspace of a product of three linearly ordered spaces, then X has the CSWP if and only if X has no subspace homeomorphic to the Cantor set. In addition, every finite power of the double arrow space has the CSWP. These results are proved using some results ...

We generalize the Bolzano-Weierstrass theorem (that every bounded sequence of reals admits a convergent subsequence) on ideal convergence. We show examples of ideals with and without the Bolzano-Weierstrass property, and give characterizations of BW property in terms of submeasures and extendability to a maximal P-ideal. We show applications to Rudin-Keisler and Rudin-Blass orderings of ideals ...

Solution: Since {xn} ∞ n=1 is a bounded sequence in R, the Bolzano-Weierstrass Theorem ensures that there is at least one convergent subsequence: let L be its limit. If the entire sequence does not converge to L, then for some > 0 there must be infinitely many terms of the sequence outside the ball B = (L− , L+ ). This infinite but bounded collection of terms likewise by Bolzano-Weierstrass has...

This paper presents a polynomial connectionist network called ridge polynomial network (RPN) that can uniformly approximate any continuous function on a compact set in multidimensional input space R (d), with arbitrary degree of accuracy. This network provides a more efficient and regular architecture compared to ordinary higher-order feedforward networks while maintaining their fast learning p...

Whether a rule-based interval type-2 fuzzy system has the ability to approximate any continuous multivariate function arbitrarily well is a fundamentally important question for fuzzy control and modeling. The only approximation results available are the preliminary ones that we previously obtained. They state that two general classes of the interval T2 fuzzy systems, one for the Mamdani type an...

Approximation theorems were formulated for function continuous in the neighborhood of some point x, -∞< x < ∞. Namely, upper bounds obtained approximations by their Gauss-Weierstrass singular operators terms a majorant first- and second-order continuity moduli relevant functions.

In this paper we find division polynomials for Huff curves, Jacobi quartics, and Jacobi intersections. These curves are alternate models for elliptic curves to the more common Weierstrass curve. Division polynomials for Weierstrass curves are well known, and the division polynomials we find are analogues for these alternate models. Using the division polynomials, we show recursive formulas for ...