Erratum to: “One-Variable and Multi-Variable Calculus on a Non-Archimedean Field Extension of the Real Numbers”

One-Variable and Multi-Mariable Calculus on a Non-Archimedean Field Extension of the Real Numbers∗

New elements of calculus on a complete real closed non-Archimedean field extension F of the real numbers R will be presented. It is known that the total disconnectedness of F in the topology induced by the order makes the usual (topological) notions of continuity and differentiability too weak to extend real calculus results to F . In this paper, we introduce new stronger concepts of continuity...

On Locally Uniformly Differentiable Functions on a Complete Non-Archimedean Ordered Field Extension of the Real Numbers

We study the properties of locally uniformly differentiable functions on N, a non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In particular, we show that locally uniformly differentiable functions are C1, they include all polynomial functions, and they are closed under addition, multiplication, and composition. Th...

One Variable Advanced Calculus

on the effect of linear & non-linear texts on students comprehension and recalling

چکیده ندارد.

Extension of the Fourier-Budan theorem to one-variable signomials

Let f(x) = a0x r0 + a1x r1 + · · · + akx rk , where each ai ∈ R, each rk ∈ N := {0, 1, . . . }, and r0 < r1 · · · < rk. Suppose u < v. Let z(f, u, v) = the number of roots of f in (u, v], counted with multiplicity. For any w ∈ R and n ∈ N, let s(f, w, n) = the number of sign-changes in the sequence f(w), f ′(w), f ′′(w), . . . , f (w) (skipping over zeros). Then the Fourier-Budan Theorem says t...