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Quantum Cognition and Quantum Brain Dynamics: Modeling Room-Temperature Coherence in Neural and Retinal Systems

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Abstract Macroscopic biological entities, such as the retina and the human brain, as well as the whole organism, are required to be continuously coherent at room temperature, which is a huge challenge for modern quantum mechanics. Biological tissue is warm, wet and noisy, and this is theoretically expected to result in rapid decoherence, but growing paradigms in quantum cognition and quantum biology indicate that the neural systems somehow avoid immediate collapse induced by environment. One of the unsolved problems in this area is the exact mechanism of how cellular structures are able to remain decoherence-free and support potentially functional quantum states. The aim of this paper is to answer this salient research gap by proposing a theoretical framework for modelling the phenomenon of quantum entanglement and non locality in the brain cell microtubules and retina photoreceptors. Combining the elements of the resource theory of quantum coherence and open quantum dynamics, we propo...
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Reconciling the Infinite Boundary: A Finite Volume Approach to Fluid Dynamics in Gabriel’s Horn

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Abstract   Gabriel's horn is a classic mathematical example of a finite volume with an infinite surface area which is also known as the Painter's Paradox. This paradox states that an infinitely large surface area would need to be coated with an infinite amount of material, in this case a paint, while it is possible to fill the interior with a finite amount of material. This paper brings together the fields of pure mathematical theory and applied computational physics by introducing a computational framework that can be used for the modeling of the hypothetical filling of Gabriel's horn with sophisticated numerical discretisation methods. Using finite volume methods (FVM) and implicit-explicit (IMEX) time integration methods, we build a hypothetical simulation pipeline to deal with the extreme geometrical tapering of the domain. The theoretical study shows that the advanced numerical schemes can effectively address the boundary interface issues in configurations that t...
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The Banach-Tarski Paradox: theoretical foundations, Axiomatic underpinnings and Methodological frameworks.

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  The Banach-Tarski Paradox: theoretical foundations, Axiomatic underpinnings and Methodological frameworks.   Abstract   One of the most surprising and counter-intuitive results in modern geometry is the Banach-Tarski paradox, which demonstrates that a ball of three dimensions can be divided into a finite number of pieces, and then rearranged into two identical replicas of the original ball. The phenomenon has a theoretical basis in the Axiom of Choice and in the existence of non-measurable sets that are a fundamental difference between pure mathematical logic and physical reality where the laws of atomism apply. In a systematic review we examine the axiomatic foundations of this theorem, explore the implications of this theorem in different mathematical spaces, and propose a hypothetical evaluation framework for equidecomposability in a discrete space, where the equidecomposability is being computed. The idea behind this is to render the apparently paradoxical d...
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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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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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Neural Network Generalization in the Over-Parameterization Regime: Mechanisms, Benefits, and Limitations

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Neural Network Generalization in the Over-Parameterization Regime: Mechanisms, Benefits, and Limitations Neural Network Generalization in the Over-Parameterization Regime: Mechanisms, Benefits, and Limitations Introduction Over the past decade, deep neural networks (DNNs) have risen to prominence across a range of machine learning applications, achieving remarkable performance in domains such as computer vision, natural language processing, and reinforcement learning. A striking and counter-intuitive feature of modern DNNs is their propensity for over-parameterization: models often contain many more parameters than training samples, far exceeding the classical regime where statistical learning theory would predict rampant overfitting and poor generalization. Yet, these highly over-parameterized models not only fit the training data perfectly but also display outstanding generalization to unseen test data—often improving as the number of paramete...
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