arXiv subjects

Davit Kapanadze

Publications and source records attributed to Davit Kapanadze.

4 recordsLinked to original sources

arXiv2026

From the Half-Order Recurrence to General Fractional-Order Differentiation on Monomials: Functional Continuation and Operator Composition

This work develops a general-order fractional differentiation operator on monomials from the discrete half-order coefficient structure established in Part I. Starting from the one-step transfer law of the integer coefficient family, the discrete coefficient relation is continued to a real argument and a canonical Gamma-functional form is selected under the stated normalization and regularity requirements. The operator order is then extended beyond one half by dividing the first derivative into equal operator steps, leading from orders 1/m and k/m to general rational and real orders. The resulting coefficient is independently verified through the addition law for operator orders on the monomial system, subject to the requirement that all intermediate expressions in the composition are defined. On the corresponding parameter domain, the construction yields the standard Gamma-ratio monomial formula for fractional differentiation. For nonnegative integer orders it reduces to ordinary repeated differentiation, while for positive non-integer orders it agrees at the monomial level with the left-sided Riemann-Liouville formula with lower limit 0. Its relation to the Caputo operator is also discussed, including the distinction for constants and low-degree polynomial terms. The aim is not to introduce a new classical fractional derivative, but to provide a constructive algebraic-operator route from a discrete coefficient law to the general fractional-order monomial formula, clarifying the roles of functional continuation, Gamma normalization, and operator composition.

math.GM↗

arXiv2026

An Algebraic-Operator Construction of the Half-Derivative on a Graded Monomial Space. Part I: Recurrence, Double Factorials, the Wallis Product, and Normalization

This paper constructs a half-order differentiation operator on the graded monomial space spanned by non-negative integer and half-integer powers of x, with x > 0. The algebraic stage uses neither an integral kernel, a limiting process, nor the Gamma function. The operator is assumed to act in the form $D^{1/2}x^β=c(β)x^{β-1/2}$. Requiring two successive applications of the operator to reproduce the ordinary first derivative yields the fundamental recurrence relation $c(β)c(β-1/2)=β$. Expanding the recurrence produces two linked coefficient families, one for integer powers and one for half-integer powers. Their formulas contain a free normalization constant $c_0=c(0)$, which cancels under composition. Thus, the compositional requirement fixes the relative coefficients but not the scale of each individual half-step. Ratios of even and odd double factorials lead naturally to a partial Wallis product; however, the Wallis product alone does not determine $c_0$. Imposing compatibility with the monomial formula for the left Riemann-Liouville half-derivative with lower limit 0 selects the value $c_0=1/\sqrtπ$. With this normalization, $D^{1/2}x^β=\frac{Γ(β+1)}{Γ(β+1/2)}x^{β-1/2}$ holds for every $β\in\{0,1/2,1,3/2,\ldots\}$. The resulting monomial formula agrees with the corresponding Riemann-Liouville formula. For positive integer powers it also agrees with the Caputo formula, whereas for the constant function it agrees only with the Riemann-Liouville rule. The resulting linear compositional operator lowers the exponent by exactly one half and is fully specified on the stated graded monomial space. Extension to broader function spaces, construction of an integral or convolution kernel, and the analysis of non-locality remain open problems.

math.GM↗

arXiv2026

A Limit-Free Algebraic-Geometric Construction of Derivatives for Elementary Functions

This paper continues the author's previous work on a limit-free algebraic-geometric construction of the derivative in the class of polynomial functions and extends the proposed framework to elementary functions. Derivatives of rational power, exponential, logarithmic, trigonometric, and inverse trigonometric functions are constructed through the geometric interpretation of the tangent line, inverse symmetry, and local linear structure, without treating the limit as the initial defining mechanism. Within the proposed approach, the derivative is introduced from the outset as a functional correspondence assigning to each point the slope coefficient of the tangent line. The paper demonstrates that the classical differentiation formulas arise naturally from interconnected geometric and algebraic structures and are subsequently consistent with standard limit-based analysis. From a methodological perspective, the study proposes the logical sequence: Tangent, Local Linear Structure, Limit formalisation. Thus, the paper presents a conceptual bridge between geometric intuition, algebraic construction, and classical differential calculus.

math.GM↗

arXiv2026

A Limit-Free Algebraic-Geometric Construction of the Derivative with a Foundational Model in the Class of Polynomial Functions

This paper presents an algebraic-geometric construction of the derivative developed initially within the class of polynomial functions without introducing limits at the initial stage. Tangency is characterized by an algebraic condition: the difference between a function and a linear approximation has a double root at a given point. On this basis, the derivative is defined as a functional correspondence assigning to each point the slope of the tangent. Within the class of polynomials, the existence, uniqueness, and fundamental rules of differentiation are established purely algebraically. The constructed model is then extended conceptually to elementary functions and connected to the linear decomposition of functions, from which the classical limit representation of the derivative naturally emerges. Thus, the limit appears not as a starting point but as an analytic expression of an already constructed concept.

math.GM↗