Fractional non-Archimedean calculus in one variable

Let ?? be a natural number. A function f: ? _p ?? K into a non-Archimedeanly valued complete field K ? ? _p is ??-times continuously differentiable if and only if its Mahler coefficients (a _n ) _n??? obey |a _n |n ^?? ?? 0 as n ?? ??. For a real number r ?? 0, this suggests the ad hoc definition by [1] of a C ^r -function f: ? _p ?? K by asking its Mahler coefficients (a _n ) _n??? to satisfy |a _n |n ^r ?? 0 as n?? ??. We will present for functions f: X ?? K on subsets X ? K without isolated points a general pointwise notion of r-fold differentiability through iterated difference quotients, subsequently shown on the domain X = ? _p to coincide with the one given above. For functions on open domains, we prove this notion to admit a handier characterization by its Taylor polynomial up to degree ?r?.     

Value-sharing and uniqueness problems for non-Archimedean differential polynomials in several variables

We consider differential polynomials of Fermat–Waring type, const-ructed using polynomials of Yi’s type for meromorphic functions in a non-Archimedean field. Similarly to the Hayman Conjecture, we prove that the considered differential polynomials assume all values. We established also a uniqueness theorem for these differential polynomials.     

Fractional calculus applied to radiation damping

Separate treatment of the low– and high–frequency part of the dynamic stiffness is essential when approximating the latter in the frequency–domain. In this paper, a doubly–asymptotic rational approximation of the low–frequency part is combined with an analytical interpretation of the asymptotic part leading to a system of fractional differential equations to represent the force–displacement relationship. Here, an analogy between fading memory of viscoelastic materials and radiation damping of unbounded domains is visible.     

Fractional calculus in viscoelasticity: An experimental study

Viscoelastic properties of soft biological tissues provide information that may be useful in medical diagnosis. Noninvasive elasticity imaging techniques, such as Magnetic Resonance Elastography (MRE), reconstruct viscoelastic material properties from dynamic displacement images. The reconstruction algorithms employed in these techniques assume a certain viscoelastic material model and the results are sensitive to the model chosen. Developing a better model for the viscoelasticity of soft tissue-like materials could improve the diagnostic capability of MRE. The well known “integer derivative” viscoelastic models of Voigt and Kelvin, and variations of them, cannot represent the more complicated rate dependency of material behavior of biological tissues over a broad spectral range. Recently the “fractional derivative” models have been investigated by a number of researchers. Fractional order models approximate the viscoelastic material behavior of materials through the corresponding fractional differential equations. This paper focuses on the tissue mimicking materials CF-11 and gelatin, and compares fractional and integer order models to describe their behavior under harmonic mechanical loading. Specifically, Rayleigh (surface) waves on CF-11 and gelatin phantoms are studied, experimentally and theoretically, in order to develop an independent test bed for assessing viscoelastic material models that will ultimately be used in MRE reconstruction algorithms.     

Electrostatics in fractal geometry: Fractional calculus approach

An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical symmetry case. The method is based on the splitting of a composite volume into a fractal volume V_d ～ r^d with the fractal dimension d and a complementary host volume V_h = V₃ ? V_d. Integrations over these fractal volumes correspond to the convolution integrals that eventually lead to the employment of the fractional integro-differentiation.     

Non-Archimedean Nevanlinna theory in several variables and the non-Archimedean Nevanlinna inverse problem

Cartan's method is used to prove a several variable, non-Archimedean, Nevanlinna Second Main Theorem for hyperplanes in projective space. The corresponding defect relation is derived, but unlike in the complex case, we show that there can only be finitely many non-zero non-Archimedean defects. We then address the non-Archimedean Nevanlinna inverse problem, by showing that given a set of defects satisfying our conditions and a corresponding set of hyperplanes in projective space, there exists a non-Archimedean analytic function with the given defects at the specified hyperplanes, and with no other defects.

     

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 ? 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 F ⁿ in general.     

Toda flows with infinitely many variables

The Toda flow and related flows extend naturally to operators in Hilbert space and the purpose of this paper is to describe these flows and to analyse some of their special properties.     

Fractional integration operators of two variables

The object of the present note is to introduce two new fractional integration operators associated with H-function of two variables. Three important properties of these operators are established which generalize the results given earlier by several authors notably by Saxena and Kumbhat,^{11, 12} Saxena and Kalla,⁵ Parashar⁸ and Lowndes.⁶     

Stability of a mixed additive and cubic functional equation in several variables in non-Archimedean spaces

Consider the functional equation ${\Im_1(f ) = \Im_2(f )\,\,(\Im)}$ in a certain general setting. A function g is an approximate solution of ${(\Im)}$ if ${\Im_1(g)}$ and ${\Im_2(g)}$ are close in some sense. The Ulam stability problem asks whether or not there is a true solution of ${(\Im)}$ near g. In this paper, we achieve the general solution and the stability of the following functional equation $$\begin{array}{ll}f\left(\sum\limits^{n}_{i=1}x_{i} \right)+f\left(\sum\limits^{n-1}_{i=1} x_{i}-x_{n} \right)\\\quad=2f\left(\sum\limits^{n-1}_{i=1}x_{i} \right)+\sum\limits^{n-1}_{i=1}(f(x_{i}+x_{n}) +f(x_{i}-x_{n})-2f(x_{i}))\end{array}$$ for all x _i (i =? 1,2, . . . , n), in non-Archimedean spaces.     

Fractional variational calculus with classical and combined Caputo derivatives

We give a proper fractional extension of the classical calculus of variations by considering variational functionals with a Lagrangian depending on a combined Caputo fractional derivative and the classical derivative. Euler-Lagrange equations to the basic and isoperimetric problems as well as transversality conditions are proved.     

Selectors in non-Archimedean spaces

Continuous selectors on the hyperspace F(X) are studied, when X is a non-Archimedean space. It is shown that a non-Archimedean space has a continuous selector if and only if it is topologically well orderable. Another characterization is given in terms of density and complete metrizability.