Khodr Shamseddine

INTERMEDIATE VALUES AND INVERSE FUNCTIONS ON NON-ARCHIMEDEAN FIELDS

Continuity or even differentiability of a function on a closed interval of a non-Archimedean field are not sufficient for the function to assume all the intermediate values, a maximum, a minimum, or a unique primitive function on the interval. These problems are due to the total disconnectedness of the field in the order topology. In this paper, we show that differentiability (in the topological sense), together with some additional mild conditions, is indeed sufficient to guarantee that the function assumes all intermediate values and has a differentiable inverse function.

Characterization of compact and self-adjoint operators on free Banach spaces of countable type over the complex Levi-Civita field

Let C be the complex Levi-Civita field and let E be a free Banach space over C of countable type. Then E is isometrically isomorphic to c0N,C,s:=(xn)n∈N:xn∈C;limn→∞|xn|s(n)=0, where s:N→0,∞. If the range of s is contained in C∖0, it is enough to study c0N,C, which will be denoted by c0(C) or, simply, c0. In this paper, we define a natural inner product on c0, which induces the sup-norm of c0. Of course, c0 is not orthomodular, so we characterize those closed subspaces of c0 with an orthonormal complement with respect to this inner product; that is, those closed subspaces M of c0 such that c0 = M ⊕ M⊥. Such a subspace, together with its orthonormal complement, defines a special kind of projection, the so-called normal projection. We present a characterization of such normal projections as well as a characterization of another kind of operators, the compact operators on c0.

Constrained second order optimization on non-archimedean fields

Constrained optimization on non-Archimedean fields is presented. We formalize the notion of a tangent plane to the surface defined by the constraints making use of an implicit function Theorem similar to its real counterpart. Then we derive necessary and sufficient conditions of second order for the existence of a local minimizer of a function subject to a set of equality and inequality constraints, based on a concept of continuity and differentiability that is stronger than the conventional one.

Generalized power series on a non-Archimedean field

Power series with rational exponents on the real numbers field and the Levi-Civita field are studied. We derive a radius of convergence for power series with rational exponents over the field of real numbers that depends on the coefficients and on the density of the exponents in the series. Then we generalize that result and study power series with rational exponents on the Levi-Civita field. A radius of convergence is established that asserts convergence under a weak topology and reduces to the conventional radius of convergence for real power series. It also asserts strong (order) convergence for points whose distance from the center is infinitely smaller than the radius of convergence. Then we study a class of functions that are given locally by power series with rational exponents, which are shown to form a commutative algebra over the Levi-Civita field; and we study the differentiability properties of such functions within their domain of convergence.

The implicit function theorem in a non-Archimedean setting*

Abstract In this paper, the inverse function theorem and the implicit function theorem in a non-Archimedean setting will be discussed. We denote by N any non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order; and we study the properties of locally uniformly differentiable functions from Nn to Nm. Then we use that concept of local uniform differentiability to formulate and prove the inverse function theorem for functions from Nn to Nn and the implicit function theorem for functions from Nn to Nm with m

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. Then we formulate and prove a version of the inverse function theorem as well as a local intermediate value theorem for these functions.

On the solutions of linear ordinary differential equations and Bessel-type special functions on the Levi-Civita field

Because of the disconnectedness of a non-Archimedean ordered field in the topology induced by the order, it is possible to have non-constant functions with zero derivatives everywhere. In fact the solution space of the differential equation y′ = 0 is infinite dimensional. In this paper, we give sufficient conditions for a function on an open subset of the Levi-Civita field to have zero derivative everywhere and we use the nonconstant zero-derivative functions to obtain non-analytic solutions of systems of linear ordinary differential equations with analytic coefficients. Then we use the results to introduce Bessel-type special functions on the Levi-Civita field and to study some of their properties.

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 and differentiability which we call derivate continuity and derivate differentiability [2, 12]; andwe show that derivate continuous and differentiable functions satisfy the usual addition, product and composition rules and that n-times derivate differentiable functions satisfy a Taylor formula with remainder similar to that of the real case. Then we generalize the definitions of derivate continuity and derivate differentiability to multivariable F-valued functions and we prove related results that are useful for doing analysis on F and Fn in general.

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

On page 160 and subsequent odd pages, in the title of the paper, the typo “Multi-Mariable” should be replaced by “Multi-Variable”.

On Integrable Delta Functions on the Levi-Civita Field

In this paper, we develop a theory of integrable delta functions on the Levi-Civita field R as well as on R2 and R3 with similar properties to the one-dimensional, two-dimensional and three-dimensional Dirac Delta functions and which reduce to them when restricted to points in R, R2 and R3, respectively. First we review the recently developed Lebesgue-like measure and integration theory over R, R2 and R3. Then we introduce delta functions on R, R2 and R3 that are integrable in the context of the aforementioned integration theory; and we study their properties and some applications.