A categorical version of the famous theorem of Stone and Weierstrass is formulated and studied in detail. Several applications and examples are given.

This note gives an alternate proof of Clairaut’s theorem—that the partial derivatives of a smooth function commute—using the Stone–Weierstrass theorem. Most calculus students have probably encountered Clairaut’s theorem. Theorem. Suppose that f : [a, b] × [c, d] → R has continuous second-order partial derivatives. Then fxy = fyx on (a, b)× (c, d). The proof found in many calculus textbooks (e.g...

Contents Chapter 1. Introduction 5 Chapter 2. Notation and preliminaries 9 1. General notation and measures 9 2. Weak * convergence of measures 10 3. Covering theorems and differentiation of measures 13 4. Hausdorff measures 13 5. Lipschitz functions 15 6. The Stone–Weierstrass Theorem 16 Chapter 3. Marstrand's Theorem and tangent measures 17 1.

A class of nonlinear Neumann problems driven by p(x)-Laplacian with a nonsmooth locally Lipschitz potential (hemivariational inequality) was considered. The approach used in this paper is the variational method for locally Lipschitz functions. More precisely, Weierstrass theorem and Mountain Pass theorem are used to prove the existence of at least two nontrivial solutions.

In their book Values of Non-Atomic Games, Aumann and Shapley [1] define the value for spaces of nonatomic games as a map from the space of games into bounded finitely additive games that satisfies a list of plausible axioms: linearity, symmetry, positivity, and efficiency. One ofthe themes of the theory of values is to demonstrate that on given spaces of games this list of plausible axioms dete...

LetX denote the rational curve with n+1 nodes obtained from the Riemann sphere by identifying 0 with∞ and ζj with −ζj for j = 0, 1, . . . , n−1, where ζ is a primitive (2n)th root of unity. We show that if n is even, then X has no smooth Weierstrass points, while if n is odd, then X has 2n smooth Weierstrass points. C. Widland [14] showed that the rational curve with three nodes obtained from P...