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Fractional-Order Bioconvection in Trihybrid Nanofluids Flowing Over a Rotating Disk: A Hybrid Neural Network With Genetic Algorithm Method for Entropy Generation Minimization

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<p>Fractional-Order Bioconvection in Trihybrid Nanofluids Flowing Over a Rotating Disk: A Hybrid Neural Network With Genetic Algorithm Method for Entropy Generation Minimization</p> : Minimizing entropy generation in complex fluid systems is a primary concern for improving thermodynamic efficiency. This paper investigates bioconvection in a Carreau-Yasuda trihybrid nanofluid over a spinning disk, where fluid memory is modeled using fractional-order derivatives. We provide an analytical energy-based stability framework for the proposed model. Given the high computational cost associated with solving fractional partial differential equations, we propose a Hybrid Neural Network surrogate model combined with a Genetic Algorithm. The Hybrid Neural Network, trained on data obtained via the Finite Difference Method, accurately predicts Nusselt numbers and entropy generation, while the Genetic Algorithm navigates the response surface to identify Pareto-optimal solutions. A deep cas...
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Prime Constellations & Base-12 Digital Roots

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Prime Constellations & Base-12 Digital Roots 🔮 Prime Constellations & Base-12 Digital Roots 🎯 What’s New? This tool identifies prime constellations—structured patterns of primes separated by fixed gaps—and filters them using base-12 digital roots . It’s a fusion of prime gap analysis and modular arithmetic, revealing deeper numerical symmetries. 💡 Base-12 Digital Root Instead of summing digits repeatedly, we use n % 11 to compute the base-12 digital root. If the remainder is 0, we treat it as 11. Valid digital roots for primes greater than 3 in base-12 are: {1, 2, 4, 5, 7, 8, 10, 11} 💻 Python Code def is_prime(n): if n < 2: return False for i in range(2, int(n**0.5)+1): if n % i == 0: return False return True def digit_root_base12(n): dr = n % 11 return dr if dr != 0 else 11 # Treat mod 11 remainder 0 as DR 11 def generate_constellations(lower, upper, gaps): valid...
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Prime Constellations & Digital Roots

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Prime Constellations & Digital Roots 🌌 Prime Constellations & Their Digital Roots 🎯 What Are Prime Constellations? Prime constellations are structured patterns of prime numbers separated by fixed gaps. Examples include: Twin Primes: (p, p+2) Cousin Primes: (p, p+4) Sexy Primes: (p, p+6) Triplets: (p, p+2, p+6) or (p, p+4, p+6) Quadruplets: (p, p+2, p+6, p+8) 💡 What Is a Digital Root? The digital root of a number is the single-digit result of repeatedly summing its digits. It helps reveal hidden numerical patterns and attractors. 💻 Python Code def is_prime(n): if n < 2: return False for i in range(2, int(n**0.5)+1): if n % i == 0: return False return True def digit_root(n): while n >= 10: n = sum(int(d) for d in str(n)) return n def generate_constellations(lower, upper, gaps): results = [] for p in range(lower, upper - ...
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Prime Constellations & Digital Roots

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Prime Constellations & Digital Roots 🌌 Prime Constellations & Their Digital Roots 🎯 What Are Prime Constellations? Prime constellations are structured patterns of prime numbers separated by fixed gaps. Examples include: Twin Primes: (p, p+2) Cousin Primes: (p, p+4) Sexy Primes: (p, p+6) Triplets: (p, p+2, p+6) or (p, p+4, p+6) Quadruplets: (p, p+2, p+6, p+8) 💡 What Is a Digital Root? The digital root of a number is the single-digit result of repeatedly summing its digits. It helps reveal hidden numerical patterns and attractors. 💻 Python Code def is_prime(n): if n < 2: return False for i in range(2, int(n**0.5)+1): if n % i == 0: return False return True def digit_root(n): while n >= 10: n = sum(int(d) for d in str(n)) return n def generate_constellations(lower, upper, gaps): results = [] for p in range(lower, upper - ...
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Sexy Primes & Digital Roots in Python

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Sexy Primes & Digital Roots in Python 💫 Sexy Primes & Their Digital Roots 🎯 What Are Sexy Primes? Sexy primes are pairs of prime numbers that differ by exactly 6. The name comes from the Latin word “sex” meaning six—not from anything risqué! Examples include (5, 11), (7, 13), and (11, 17). 💡 What Is a Digital Root? The digital root of a number is the single-digit value obtained by repeatedly summing its digits until only one digit remains. For example: 137 → 1 + 3 + 7 = 11 → 1 + 1 = 2 89 → 8 + 9 = 17 → 1 + 7 = 8 💻 Python Code def is_prime(n): if n < 2: return False for i in range(2, int(n**0.5) + 1): if n % i == 0: return False return True def digit_root(n): while n >= 10: n = sum(int(d) for d in str(n)) return n def find_sexy_primes_with_roots(lower, upper): sexy_pairs = [] for p in range(lower, upper - 6): if is_prime(p) and is...
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Cousin Primes & Digital Roots: Interactive Python Tool

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Cousin Primes & Digital Roots: Interactive Python Tool 🔍 Cousin Primes & Their Digital Roots 🎯 What Are Cousin Primes? Cousin primes are pairs of prime numbers that differ by exactly 4. Examples include (3, 7), (7, 11), and (13, 17). These pairs help us explore prime gaps and distribution patterns in number theory. 💡 What Is a Digital Root? The digital root of a number is the single-digit value obtained by repeatedly summing its digits until only one digit remains. For example: 137 → 1 + 3 + 7 = 11 → 1 + 1 = 2 89 → 8 + 9 = 17 → 1 + 7 = 8 💻 Python Code def is_prime(n): if n < 2: return False for i in range(2, int(n**0.5) + 1): if n % i == 0: return False return True def digit_root(n): while n >= 10: n = sum(int(d) for d in str(n)) return n def find_cousin_primes_with_roots(lower, upper): cousin_pairs = [] for p in range(lower, upper - 4):...
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Cousin Primes & Digital Roots in Python

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Cousin Primes & Digital Roots in Python 🔍 Cousin Primes & Their Digital Roots 🎯 What Are Cousin Primes? Cousin primes are pairs of prime numbers that differ by exactly 4. Examples include (3, 7), (7, 11), and (13, 17). These pairs offer insights into prime gaps and distribution patterns. 💡 What Is a Digital Root? The digital root of a number is the single-digit value obtained by repeatedly summing its digits until only one digit remains. For example: Digit root of 137 → 1 + 3 + 7 = 11 → 1 + 1 = 2 Digit root of 89 → 8 + 9 = 17 → 1 + 7 = 8 💻 Python Code def is_prime(n): if n < 2: return False for i in range(2, int(n**0.5) + 1): if n % i == 0: return False return True def digit_root(n): while n >= 10: n = sum(int(d) for d in str(n)) return n def find_cousin_primes_with_roots(limit): cousin_pairs = [] for p in range(2, limit - 4): if...
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Interactive Cousin Prime Finder in Python

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Interactive Cousin Prime Finder in Python 🧮 Find Cousin Primes with Python 🎯 What Are Cousin Primes? Cousin primes are pairs of prime numbers that differ by exactly 4. Examples include (3, 7), (7, 11), and (13, 17). These pairs are part of the study of prime gaps and help us understand how primes are distributed. 💡 Interactive Approach This Python script allows users to input an upper limit and dynamically find all cousin prime pairs up to that number. 💻 Python Code def is_prime(n): if n < 2: return False for i in range(2, int(n**0.5) + 1): if n % i == 0: return False return True def find_cousin_primes(limit): cousin_pairs = [] for p in range(2, limit - 4): if is_prime(p) and is_prime(p + 4): cousin_pairs.append((p, p + 4)) return cousin_pairs def main(): try: user_limit = int(input("Enter the upper limit to find cousin primes: ")) ...
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Free Field Operator: Building Quantum Fields

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Free Field Operator: Building Quantum Fields How Quantum Fields Evolve Without Interactions 🎯 Our Goal We aim to construct the free scalar field operator \( A(x,t) \), which describes a quantum field with no interactions—just free particles moving across space-time. 🧠 Starting Expression This is the mathematical formula for our field \( A(x,t) \): \[ A(x, t) = \frac{1}{(2\pi)^{3/2}} \int_{\mathbb{R}^3} \frac{1}{\sqrt{k_0}} \left[ e^{i(k \cdot x - k_0 t)} a(k) + e^{-i(k \cdot x - k_0 t)} a^\dagger(k) \right] \, dk \] x: Spatial position t: Time k: Momentum vector k₀ = √(k² + m²): Relativistic energy of the particle a(k): Operator that removes a particle (annihilation) a†(k): Operator that adds a particle (creation) 🧩 What Does This Mean? The field is made up of wave patterns (Fourier modes) linked to momentum \( k \). It behaves like a system that decides when and where ...
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Understanding the Laplacian of 1/r and the Dirac Delta Function Mathematical Foundations & SageMath Insights

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Heuristic Computation and the Discovery of Mersenne Primes

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Heuristic Computation and the Discovery of Mersenne Primes Heuristic Computation and the Discovery of Mersenne Primes “Where Strategy Meets Infinity: The Quest for Mersenne Primes” Introduction: The Dance of Numbers and Heuristics Mersenne primes are not just numbers—they are milestones in the vast landscape of mathematics. Defined by the formula: \[ M_p = 2^p - 1 \] where \( p \) is itself prime, these giants challenge our computational limits and inspire new methods of discovery. But why are these primes so elusive? As \( p \) grows, the numbers become astronomically large, making brute-force testing impossible. This is where heuristic computation steps in—guiding us with smart, experience-driven strategies. “In the infinite sea of numbers, heuristics are our compass.” Let’s explore how heuristics and algorithms intertwine to unveil these mathematical treasures. 1. Mersenne Primes — Giants of Number Theory Definition: Numbers of the form \( M_p = 2^p - 1 \...
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