Related papers: Calculus on Dual Real Numbers
We use Taylor's formula with Lagrange remainder to make a modern adaptation of Poisson's proof of a version of the fundamental theorem of calculus in the case when the integral is defined by Euler sums, that is Riemann sums with left (or…
We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for the Bernoulli and Euler numbers and polynomials.
We show that the Schubert calculus of enumerative geometry is real, for special Schubert conditions. That is, for any such enumerative problem, there exist real conditions for which all the a priori complex solutions are real.
We look at the fundamental use of complex numbers in Quantum Mechanics (QM). A review of some of the most popular reasons given in the literature to support the necessity of the complex formalism, We add some insight by invoking others.…
A constructive approach to differential calculus on quantum principal bundles is presented. The calculus on the bundle is built in an intrinsic manner, starting from given graded (differential) *-algebras representing horizontal forms on…
Given any polynomial with real coefficients, the existence of a real quadratic polynomial factor is proven using only basic real analysis. The aim is to provide an approachable proof to anybody who is familiar with the least upper bound…
The aim of this work is to derive a symbol calculus on $L^2(\mathbb{R})$ for one-dimensional Hausdorff operators in apparently the most general form.
In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…
Duality between the coloured quantum group and the coloured quantum algebra corresponding to GL(2) is established. The coloured L^{\pm} functionals are constructed and the dual algebra is derived explicitly. These functionals are then…
In the present article, modeling certain rational numbers, that are represented in terms of Cantor series, are described. The statements on relations between digits in the representations of rational numbers by Cantor series (for the case…
By using Alexander duality on simplicial complexes we give a new and algebraic proof of Dirac's theorem on chordal graphs.
A term calculus for the proofs in multiplicative-additive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive…
We give a counting based proof of the Graham Pollak Theorem
This note shows that the three theorems presented in J. Math. Anal. Appl. 556 (2026), 130199, whose proofs, in their present formulation, are purely formal, follow from elementary calculus.
We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the first n digits of its continued fraction expansion performs in the order of n^4 mathematical operations. The…
A short introduction to the mathematical methods and technics of differential algebras and modules adapted to the problems of mathematical and theoretical physics is presented.
A description of solutions of some integral equations has been obtained. A two-radii theorem is obtained as well.
This thesis is intended to provide an account of the theory and applications of Operational Methods that allow the "translation" of the theory of special functions and polynomials into a "different" mathematical language. The language we…
The proofs that the real numbers are denumerable will be shown, i.e., that there exists one-to-one correspondence between the natural numbers $N$ and the real numbers $\Re$. The general element of the sequence that contains all real numbers…
This article seeks to encourage a mathematical dialog regarding a possible solution to Beals Conjecture. It breaks down one of the worlds most difficult math problems into laymans terms and encourages people to question some of the most…