Hemos desarrollado un enfoque combinatorio fundamentado en el Triángulo de Pascal para abordar el problema clásico de las sumas de potencias de números naturales. Este método se basa en tres secuencias clave derivadas del triángulo: los coeficientes binomiales, los números de Narayana y una variante de los coeficientes binomiales. A través de estas secuencias, hemos formulado tres estrategias sistemáticas: el Método Directo (M.D), el Método de Coeficientes (M.C) y el Análisis de Coeficientes (A.C), las cuales potencian tanto la aritmética como el reconocimiento de patrones numéricos, alcanzando resultados equivalentes a los obtenidos mediante enfoques puramente algebraicos, como las sumas telescópicas.

Hemos desarrollado un enfoque combinatorio fundamentado en el Triángulo de Pascal para abordar el problema clásico de las sumas de potencias de números naturales. Este método se basa en tres secuencias clave derivadas del triángulo: los coeficientes binomiales, los números de Narayana y una variante de los coeficientes binomiales. A través de estas secuencias, hemos formulado tres estrategias sistemáticas: el Método Directo (M.D), el Método de Coeficientes (M.C) y el Análisis de Coeficientes (A.C), las cuales potencian tanto la aritmética como el reconocimiento de patrones numéricos, alcanzando resultados equivalentes a los obtenidos mediante enfoques puramente algebraicos, como las sumas telescópicas.

Integrals play a fundamental role in various scientific and industrial applications. Their significance extends beyond theoretical mathematics to practical fields such as physics, chemistry, and engineering. One notable application is in volume determination, which is essential for industries designing product containers. This project explores the use of integrals to derive a general expression for the volume of a Pilsen bottle using the Lagrange method. The motivation for this study arose from a simple yet intriguing question: why are certain beverage bottles made of glass despite their fragility? To answer this, we investigate how integral calculus, particularly the disk method and proportionality properties, can be applied to model the bottle’s shape and compute its volume. By integrating mathematical techniques with real-world applications, this research demonstrates the practical utility of integrals in industrial design and production.

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The aim of this work is to optimize the existing formula based on Wilson's theorem to reduce the magnitude of the computation results. Wilson's theorem states: if p is a prime number, then (p-1)! 1 is divisible by p (p-1)! ≡ -1 (mod p). The function (p-1)! increases very rapidly and reaches huge values. When the values of p are large, the calculations become resource-intensive, so it is necessary to reduce the upper limit of the calculation results.

The article introduces very simple and quite effective algorithm for constant state detection in time series. The algorithm, based on sliding window of variable length, searches a sections of time series with some given minimal length, that have all the values in some given range. It is shown that the computational complexity of afore- mentioned algorithm is O(N log N), where N is the length of time series.

The purpose of this work is to obtain exact and approximate formulas that calculate the number of partitions of even numbers into sums of pairs of prime numbers.

Introduces a novel computational approach for efficiently navigating infinite multi-dimensional spaces using a bellande step function within a model-integrated dimensional framework. By optimizing the step function, the method addresses challenges in high-dimensional path-finding and node calculation, allowing movement towards target nodes within defined distance constraints. Leveraging this approach, efficiently compute the next step towards a target node, ensuring accurate movement while adhering to specified distance limits. The integration of dimensional space modeling enhances the path-finding process, demonstrating improved computational efficiency and accuracy. The results underscore the robustness and scalability of this approach, showcasing its potential applications in robotics, path-finding, and complex systems modeling. This integration of the step function with model-integrated dimensional space represents a significant advancement in the computational efficiency and precision of multi-dimensional node calculations.

Let X be a compact Riemann surface of genus g ≥ 2 and M(E6) be the moduli space of E6-Higgs bundles over X. We consider the automorphisms σ+ of M(E6) defined by σ (E, φ) = (E∗ , −φt) , induced by the action of the outer involution of E6 in M(E6), and σ− defined by σ (E, φ) = (E* , -φt), which results from the combination of σ+ with the involution of M(E6), which consists on a change of sign in the Higgs field. In this work, we describe the fixed points of σ+ and σ−, as F4-Higgs bundles, F4-Higgs pairs associated with the fundamental irreducible representation of F4, and PSp(8, C)-Higgs pairs associated with the second symmetric power or the second wedge power of the fundamental representation of Sp(8, C). Finally, we describe the reduced notions of semistability and polystability for these objects. Citation: Antón-Sancho, Álvaro. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface. Open Mathematics, vol. 20, no. 1, 2022, pp. 1723-1733. https://doi.org/10.1515/math-2022-0543

This refers to the process of expressing the square of the real part of complex numbers as a product of its factors. In mathematical terms, if you have a complex number z = a biz=a+bi, where a-a is the real part and bb is the imaginary part, the factorization involves breaking down {Re}(z))^2(Re(z)) into its constituent factors. This process can be useful in various mathematical contexts, such as simplifying expressions.