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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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Taming the Infinite: Singularities, Regularization, and Analytic Continuation Explained

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Taming the Infinite: Singularities, Regularization, and Analytic Continuation Explained Matrix Space Toolkit in SageMath Taming the Infinite – How We Make Sense of "Impossible" Functions! Introduction: The Mystery of the Infinite (and Why We Care!) What happens when a function tries to break mathematics? Can we ever truly understand something that goes to infinity? These aren’t just philosophical musings. In both pure math and applied science, functions that misbehave—spiking to infinity or becoming undefined—are everywhere. And yet, they’re essential. But how do we work with something that shouldn’t be computable? Take 1/x. It's fine—until you hit x = 0, where it suddenly becomes undefined. This is what mathematicians c...
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Unlocking Distribution Theory: Understanding Generalized Functions & derivatives

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Unlocking Distribution Theory: Understanding Generalized Functions & Derivatives Matrix Space Toolkit in SageMath From Smooth Functions to Distributions: What Happens When You Differentiate a Functional? Introduction: More Than Just Derivatives If you've ever taken a calculus class, you know how to differentiate a function. But what if you're not differentiating a function—but a functional? Even more mind-bending: what if the object you're working with isn't even a function in the traditional sense, but a generalized function or distribution? Welcome to the magical world of distribution theory, where even the Dirac delta "function" makes perfect sense, and derivatives can be defined in a way that bypasses a...
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Generalized Functions & Differential Equations: Exploring the Infinite & the Unexpected

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Understanding Delta Function Approximations: Lorentzian, Gaussian, and Sinc Compared Matrix Space Toolkit in SageMath Differential Equations for Generalized Functions: When Calculus Meets the Infinite and the Weird In the previous blog, we understood the Understanding Delta Function Approximations: Lorentzian, Gaussian, and Sinc Compared . Let's take another one step and explore the Differential Equations for Generalized Functions: What happen When Calculus Meets the Infinite and the Weird What happens when you mix the familiar world of differential equations with the strange universe of generalized functions—those magical creatures that extend what we usually call a function? It turns out, you get a whole new playground where class...
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Understanding Delta Function Approximations: Lorentzian, Gaussian, and Sinc Compared

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Understanding Delta Function Approximations: Lorentzian, Gaussian, and Sinc Compared Matrix Space Toolkit in SageMath Delta-Convergent Sequences — Refined Blog with SageMath Symbolics, Physics Insights, and Cleaner Code In the previous blog, we understood the Lorentzian Delta Sequence (Cauchy Kernel) , Gaussian Approximation (Heat Kernel) and Sinc Delta Sequence . Let's take another one step and explore the Understanding Delta Function Approximations: Lorentzian, Gaussian, and Sinc Compared. Why Study These Approximations? Delta functions are central in many fields: Signal Processing: Ideal impulse, filter response Physics: Point charges/masses, Green's functions Spectral Theory: Lorentzian prof...
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Understanding Delta Function Approximations: Sinc-Based Approximation (Fourier Kernel)

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Understanding Delta Function Approximations: Sinc-Based Approximation (Fourier Kernel) Matrix Space Toolkit in SageMath Delta-Convergent Sequences — Refined Blog with SageMath Symbolics, Physics Insights, and Cleaner Code In the previous blog, we understood the Lorentzian Delta Sequence (Cauchy Kernel) and Gaussian Approximation (Heat Kernel) . Let's take another one step and explore the Sinc-Based Approximation (Fourier Kernel). Why Study These Approximations? Delta functions are central in many fields: Signal Processing: Ideal impulse, filter response Physics: Point charges/masses, Green's functions Spectral Theory: Lorentzian profiles in resonance Diffusion Models: Gaussians arise from ...
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Understanding Delta Function Approximations: Gaussian Delta Sequence (Heat Kernel)

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Understanding Delta Function Approximations: Gaussian Delta Sequence (Heat Kernel) Matrix Space Toolkit in SageMath Delta-Convergent Sequences — Refined Blog with SageMath Symbolics, Physics Insights, and Cleaner Code In the previous blog, we understood the Lorentzian Delta Sequence (Cauchy Kernel) . Let's take another step and explore the Gaussian Delta Sequence (Heat Kernel). Why Study These Approximations? Delta functions are central in many fields: Signal Processing: Ideal impulse, filter response Physics: Point charges/masses, Green's functions Spectral Theory: Lorentzian profiles in resonance Diffusion Models: Gaussians arise from the heat equation Numerics: Regularizing singular...
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Understanding Delta Function Approximations: Lorentzian Delta Sequence (Cauchy Kernel)

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Understanding Delta Function Approximations: Lorentzian Delta Sequence Matrix Space Toolkit in SageMath Delta-Convergent Sequences — Refined Blog with SageMath Symbolics, Physics Insights, and Cleaner Code The Dirac delta function isn’t a “normal” function — it’s an idealization used to represent a point source. It's infinitely narrow, infinitely tall, and yet integrates to 1. We approximate it using delta-convergent sequences: real functions depending on a parameter that becomes increasingly peaked at zero as the parameter vanishes. This post explores the three most common delta-approximating sequences using SageMath, including plots, integration checks, and real-world meaning. Why Study These Approximations? Delta function...
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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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Unmasking the Laplacian: How Mathematics, Physics & AI Use This Powerful Operator Unveiling the Laplacian's Secrets: A Look at 1/r with SageMath Engage & Explore! Before we dive into the math, ask yourself: What does it mean when a function explodes to infinity at a point? Leave your thoughts in the comments below. We'll revisit this after exploring the Dirac delta function! Why Is the Laplacian of \( \frac{1}{r} \) Important? The function \( \frac{1}{r} \) frequently appears in physics to describe potentials like gravity and electrostatics, which depend inversely on distance. Understanding its Laplacian reveals the nature of sources concentrated at singular points. Mathematical Foundation: What Is the Laplacian and a Harmonic Function? The Laplacian operat...
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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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