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How Gaussian Distribution Maximizes Entropy — The Proof
How Gaussian Distribution Maximizes Entropy — The Proof
The normal distribution, often referred to as the Gaussian distribution, stands as a fundamental element in the realm of statistics and…
Freedom Preetham
May 4
Mathematical Bridge Between Softmax Functions and Gibbs Distributions
Mathematical Bridge Between Softmax Functions and Gibbs Distributions
The softmax function is an essential element in various neural networks, particularly those designed for classification tasks. It…
Freedom Preetham
Apr 25
Mathematical Insights into Multi-Class Cross-Entropy Loss
Mathematical Insights into Multi-Class Cross-Entropy Loss
The multi-class cross-entropy loss is an extension of the concept of entropy in information theory, applied to the context of machine…
Freedom Preetham
Apr 25
The Gaussian Distribution: A Mathematical Perspective
The Gaussian Distribution: A Mathematical Perspective
The normal distribution, often referred to as the Gaussian distribution, is pivotal in statistics, owing to its fundamental mathematical…
Freedom Preetham
Apr 24
Tensor Contraction and Dimensionality Reduction
Tensor Contraction and Dimensionality Reduction
Tensor contraction is a nuanced and potent mathematical operation employed extensively across various scientific disciplines, including…
Freedom Preetham
Apr 14
The Mathematical Foundation to Understand Human Genomic Variation
The Mathematical Foundation to Understand Human Genomic Variation
Despite 99.9% genetic similarity among humans, the phenotypic diversity we observe — ranging from physical appearance to disease…
Freedom Preetham
Apr 12
A Mathematical Exploration of the Computational Universe: A Critique of Wolfram’s Framework
A Mathematical Exploration of the Computational Universe: A Critique of Wolfram’s Framework
The concept of a computational universe as proposed by Stephen Wolfram posits that the universe’s complexity arises from simple…
Freedom Preetham
Nov 9, 2023
Part 6 — Tensor Algebra: Dissecting Multidimensional Genomic Interactions
Part 6 — Tensor Algebra: Dissecting Multidimensional Genomic Interactions
As we proceed in our voyage through the genomic universe, the spotlight now turns to Tensor Algebra, a potent mathematical framework that…
Freedom Preetham
Nov 5, 2023
Part 5 — Statistical Mechanics: Probing the Dynamic Behavior in Genomic Systems
Part 5 — Statistical Mechanics: Probing the Dynamic Behavior in Genomic Systems
The voyage into the genomic universe continues as we harness the powers of statistical mechanics to decipher the dynamic behavior…
Freedom Preetham
Nov 5, 2023
Differential Geometry: Unveiling the Geometric Structure of Genomic Grammar — Part 4
Differential Geometry: Unveiling the Geometric Structure of Genomic Grammar — Part 4
The venture into the genomic landscape further deepens as we employ the tools of differential geometry to delineate the geometric…
Freedom Preetham
Nov 3, 2023
Algebraic Topology: Charting the Topological Landscape of Genomic Grammar — Part 3
Algebraic Topology: Charting the Topological Landscape of Genomic Grammar — Part 3
The realm of genomics is a vibrant space of intricate interactions and multifaceted regulatory frameworks. The application of algebraic…
Freedom Preetham
Nov 3, 2023
Tensor Representations: Understanding the N-Dimensional Genomic Grammar — Part 2
Tensor Representations: Understanding the N-Dimensional Genomic Grammar — Part 2
Building upon the foundational discussion laid out in Part 1, “A Rigorous Mathematical Exposition on N-Dimensional Genomic Grammar vs…
Freedom Preetham
Nov 2, 2023
Part 1 — A Rigorous Mathematical Exposition on N-Dimensional Genomic Grammar vs One-Dimensional…
Part 1 — A Rigorous Mathematical Exposition on N-Dimensional Genomic Grammar vs One-Dimensional…
People ask why can’t we just point an LLM architecture to a DNA sequence and just decode the language of life! Just like how we did for…
Freedom Preetham
Nov 2, 2023
Mathematical Explanation of Chaos, Randomness, and Complexity in the Universe.
Mathematical Explanation of Chaos, Randomness, and Complexity in the Universe.
Freedom Preetham
Oct 1, 2023
Prime Mysteries: The Logarithmic Integral and Analytic Theory
Prime Mysteries: The Logarithmic Integral and Analytic Theory
Prime numbers have always stood as sentinels in the mathematical realm — ancient, mysterious, and foundational. Their distribution across…
Freedom Preetham
Sep 24, 2023
Understanding the 1D Fractional Laplacian Equation
Understanding the 1D Fractional Laplacian Equation
Differential equations often serve as the backbone of many scientific models. While many are familiar with the traditional differential…
Freedom Preetham
Sep 9, 2023
Optimizing AI Platform Pricing Using the Lagrange Multiplier Method
Optimizing AI Platform Pricing Using the Lagrange Multiplier Method
In the competitive landscape of Artificial Intelligence platform services, pricing plays a pivotal role in determining a product’s success…
Freedom Preetham
Aug 30, 2023
Math Paper: Graph Gradient Diffusion
Math Paper: Graph Gradient Diffusion
As part of AI research, one of my main objectives is to do research in math, especially in Dynamical Systems. While I was looking for…
Freedom Preetham
Aug 20, 2023
Demystifying the Jacobian: An Essential Tool in Multivariable Calculus
Demystifying the Jacobian: An Essential Tool in Multivariable Calculus
You may have encountered the term ‘Jacobian’ during your explorations of multivariable calculus. Personally, when I first stumbled upon it…
Freedom Preetham
Jul 9, 2023
The Productivity Paradox in the AI Era: An Econometric Exploration
The Productivity Paradox in the AI Era: An Econometric Exploration
An intriguing dilemma looms over the brave new world promised by artificial intelligence (AI) and machine learning: despite significant…
Freedom Preetham
Jul 5, 2023
Quintic Unsolvability
Quintic Unsolvability
If you are in computer science, physics, or any applied math discipline, you would have learned the numerical factorization of polynomials…
Freedom Preetham
Oct 30, 2022
Confidence, Tolerance, and Prediction Intervals for Statistical Forecasts.
Confidence, Tolerance, and Prediction Intervals for Statistical Forecasts.
When dealing with statistical forecasting, you hear three specific types of intervals. The confidence interval, tolerance interval, and…
Freedom Preetham
Oct 23, 2022
Estimating Correlations through Constant Rate or Monotonicity. But what is the Question?
Estimating Correlations through Constant Rate or Monotonicity. But what is the Question?
When you come across correlation relationships, you quite often stumble upon wanting to know how your independent variable is affecting…
Freedom Preetham
Oct 19, 2022
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