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| viXra.org > General Mathematics > 2605 |
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[6] viXra:2605.0086 [pdf] submitted on 2026-05-20 22:18:45
Authors: Giuseppe D'Ambrosio
Comments: 20 Pages. (Note by viXra Admin: Author name is required in the article after the article title)
This paper introduces the D’Ambrosio Integral, a geometrically oriented generalization of the Riemann integral that incorporates the curvature of the integrable curve. Unlike classical schemes based on linear approximation, the proposed method employs circular arcs determined by consecutive triplets of points, allowing curvature information to be directly embedded into the integration process. It is shown that the integral converges, is geometrically invariant, and coincides with the Riemann integral for sufficiently smooth functions. The formulation naturally extends to Riemannian manifolds through the use of the exponential map. The proposed approach provides higher geometric accuracy in regions of pronounced curvature and is applicable to the analysis of curves, vector fields, and dynamical systems.
Category: General Mathematics
[5] viXra:2605.0082 [pdf] submitted on 2026-05-19 23:28:32
Authors: Tue Vu, Trista Vu
Comments: 248 Pages. [License] CC-BY-NC 4.0
As we enter Part II, we turn to two new subjects: Extensions of Fundamental Trigonometric and Hyperbolic Functions and Unraveling the Mystery of the Riemann Hypothesis: Toward a Complete Proof, beginning with Chapter 3 and Chapter 4.Chapter 3 develops generalized forms of classical trigonometric and hyperbolic functions, elevating them to a broader conceptual framework. These extended formulations introduce new dynamic behaviors, enabling the reshaping of familiar functions and the exploration of entirely new families of elementary functions. This extension creates a richer mathematical framework for studying natural curved surfaces. Classical trigonometric functions such as sin(x) and cos(x) produce identical cross sections when plotted in three dimensions, yielding surfaces that replicate the same curve for every value of y. In contrast, the new two variable trigonometric functions generate curves that vary with y, producing surfaces that differ across the domain and giving rise to distinct geometric objects rather than simple extrusions.Chapter 4 presents a proposed proof of the Riemann Hypothesis, one of the most intriguing and significant problems in mathematics. We explore known findings and properties of the Riemann zeta function, reformulate the functional equation in a new light, and demonstrate why certain hypothetical pairs of nontrivial zeros cannot exist. We present a proof of the Riemann Hypothesis using an elementary algebraic approach. In Section 4-4, we demonstrate that all possible nontrivial zeros lie in the narrow bands 6.28318534... ≤ 6.28983598... or -6.28983598... ≤ Im(s) <-6.28318534... for 0 However, none of them serve as solutions to ζ(s) = 0 because their imaginary parts are below the first nontrivial zero. This approach aims to clarify the assertion that all nontrivial zeros of the zeta function lie on the vertical line where the real part is 1/2.
Category: General Mathematics
[4] viXra:2605.0058 [pdf] submitted on 2026-05-15 08:53:12
Authors: Zhi Li, Hua Li
Comments: 10 Pages.
This paper discovers and proves the existence of a general, non-radical-based formulaic solution for the general quintic equation with complex coefficients. The method involves transforming the general form of the equation into a formulaically solvable form—referred to herein as standard form— given by x^5−px+ 1 = 0 , where p is an arbitrary complex number. When the modulus of p satisfies |p|≥1.65, the solution is derived using a series expansion involving negative integer powers with its coefficients of an integral series; conversely, when |p|< 1.65 , a series expansion involving positive powers with its coefficients of a fractional series is employed. These two approaches form a complete and logically closed loop. This method is purely algebraic in nature, requiring neither root searching nor iterative procedures. Furthermore, since any general quintic equation with complex coefficients can invariably be transformed into this standard form, the proposed method possesses universal applicability. Numerical results demonstrate that the method presented in this paper is highly practical and easy to implement.
Category: General Mathematics
[3] viXra:2605.0049 [pdf] submitted on 2026-05-13 10:39:53
Authors: Timothy Jones
Comments: 4 Pages.
The TI84 has both a definite integral function and a recursive list generator. We explore whether the combination can be used to solve single and double integral problems that reference recursive formulas for integral evaluations.
Category: General Mathematics
[2] viXra:2605.0044 [pdf] submitted on 2026-05-12 20:44:05
Authors: Abdoul Kader Adamou
Comments: 96 Pages. In French; Licence: CC BY-SA 4.0 (Note by viXra.org Admin: Please cite listed scientific references)
This new word "matregraph" comes from the combination of two terms already known in the field of mathematics which are: matrix and graph. A "matregraph" is precisely u200bu200ba new mathematical tool with several functions that assemble, rank and perform arithmetical calculations between real numbers (IR) in order to reach to a general result located in the structure of this "matregraph" at the place of n-line and n-column. It’s an algorithm that is able to select some values among a lot of values reducing consequently a long calculation by getting quickly a result. This new theory may be used in many fields such as physics, chemistry, computer science, telecommunications, electricity, cryptography, mechanics, transportation ...etc. Finally, "matregraph" is u200bu200ba mathematical model that is flexible and may help us solve many problems of science and engineering.
Category: General Mathematics
[1] viXra:2605.0043 [pdf] submitted on 2026-05-12 20:40:57
Authors: Abdoul Kader Adamou
Comments: 36 Pages. In French; Licence: CC BY-SA 4.0 (Note by viXra.org Admin: Please cite listed scientific references)
Natural numbers used here in mechanics, topology and quantum context is a new branch of modern mathematics whose deep abstraction and revolutionary nature make a great change in our underderstanding of mathematics field’s fundamental knowledge. This new theory applies principles from mechanics, topology and quantum field to natural numbers. It invites us to explore a new type of reasoning where we will use new calculation methods with innovative tools that are useful for mathematical research, particularly for complex open problems and likely useful for other fields of science and technology.
Category: General Mathematics
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