| viXra.org > Functions and Analysis > 2512 |
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| viXra.org > Functions and Analysis > 2512 |
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[2] viXra:2512.0101 [pdf] submitted on 2025-12-21 23:29:17
Authors: Teo Banica
Comments: 400 Pages.
This is an introduction to the study of real functions, $f:mathbb Rtomathbb R$. We first discuss motivations and examples, ways of representing functions, and with a detailed look into the basic functions, namely polynomials, and $sin,cos,exp,log$. Then we discuss continuity, with the standard results on the subject, and notably with the Weierstrass approximation theorem. We then discuss derivatives, again with the standard results on the subject, notably with the Taylor formula and its applications. Finally, we discuss integrals, with what can be done with Riemann sums, the relation with the derivatives, and with a look into more advanced functional analysis, and several variables too.
Category: Functions and Analysis
[1] viXra:2512.0008 [pdf] submitted on 2025-12-03 22:08:07
Authors: Alexander Pisani
Comments: 23 Pages. (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org)
This paper establishes a rigorous isomorphism between probability mathematics and informational resistance. The fundamental translation Ω(P) = −ln(P) maps probability to resistance, transforming multiplication into addition. This correspondence extends across four equivalent representations (natural language, probability theory, circuit topology, and prime coordinate vectors) which function as a Rosetta Stone for discrete mathematics: the same computation can be expressed in any layer and translated exactly to the others.Boolean logic emerges from circuit topology: AND as series (resistances add), OR as parallel (conductances add), and NOT as phase interference. We prove the framework's validity through two complete worked examples. First, we derive the probability that two random integers are coprime, obtaining the known result 6/π² by traversing all four representation layers. Second, we demonstrate that relational database operations (JOIN, UNION, INTERSECTION) map directly to number-theoretic operations (GCD, LCM). We establish Turing completeness by constructing an explicit simulation of counter machines using prime exponents as registers. The framework reveals that prime numbers are precisely the irreducible elements of this informational structure, configurations whose resistance cannot be decomposed into sums of smaller resistances. This characterization, combined with the Gödel-style encoding of data through prime coordinates, suggests that any sufficiently powerful computational system must rediscover the primes as a structural necessity.
Category: Functions and Analysis
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