ENDOCRINOPATHY OR EARLY PUBERYNY: NUTRITIONAL AND CHEMICAL ASSESSMENT OF PACKAGED FOOD PRODUCTS IN CHILDREN

-
ENDOCRINOPATHY OR EARLY PUBERYNY: NUTRITIONAL AND CHEMICAL ASSESSMENT OF PACKAGED FOOD PRODUCTS IN CHILDREN Abstract The global rise in early puberty in children is an important public health problem, which needs a multi-disciplinary, toxicological, nutritional and computational study. Packaged foods are most common foods consumed by children in today's diet and are a double-edged sword as hyper palatable foods containing excess amounts of sugar and caloric density, and simultaneously containing a hidden vector of exposure to endocrine disrupting chemicals (EDCs) via the synthetic packaging materials. This multi-faceted issue is addressed by a novel, comprehensive method that couples the use of AI-based dietary assessments with quantitative structure activity relationships (QSAR) toxicological modelling and an efficient Bayesian ordinal quantile regression model to dissect complex developmental endpoints. The framework allows for high fidelity exposure information as the packag...
Post a Comment

Advanced Integration Techniques with SageMath: Visual Guides, Riemann Sums, Step-by-Step Examples, and Real-World Applications(Part 2)

-

 

🔢 Riemann Sums Demystified: Left, Right, and Midpoint Rules with SageMath

Ready to turn curves into sums and slices into insights? In this post, we’ll uncover how different Riemann sum techniques help us approximate integrals — one rectangle at a time! 🧮📏

🧭 Quick Recap: Why Riemann Sums?

Riemann sums are a foundational way to estimate definite integrals — great for understanding accumulation when exact methods are tough. They're used everywhere from physics simulations to biological modeling and even digital graphics!

◀️ 1. The Left Riemann Sum

This method uses the left endpoint of each subinterval. It’s simple and quick — a great starting point!

🔍 Try this:

Can you change the function to f(x) = sin(x) and see how the result compares?

▶️ 2. The Right Riemann Sum

This version uses the right endpoint of each interval. It often gives a better estimate when the function is increasing.

📊 Notice how the estimate shifts compared to the left sum!

⏺️ 3. The Midpoint Rule

The midpoint often gives the best approximation — a balance between underestimation and overestimation.

🎨 Interactive Visualization

Experiment live with SageMath! Use the SageMathCell widget below to play around with Riemann sums and watch how the method and number of intervals affect the result. Click on the widget to explore different functions, methods, and subintervals in real-time. Try it here!

✨ Reflect & Explore

  • Which method gives the closest approximation for your chosen function?
  • What happens when you increase or decrease the number of subintervals?
  • Can you think of a physical situation (like estimating fuel usage or distance traveled) where these sums might be useful?

🔗 Dive Deeper with SageMath

  • SageMath Documentation on Numerical Integration
  • Try SageMathCell Online

💬 Let’s Collaborate!

Tried a cool function or noticed something interesting? Share your results, observations, or even a real-world application of Riemann sums in the comments below! Let’s learn from each other and make this a fun, interactive space. 🚀

🔜 Up Next: The Fundamental Theorem of Calculus!

We’ve learned how Riemann sums help approximate integrals — now let’s see how the Fundamental Theorem of Calculus ties everything together! 📚✨

Stay tuned — this theorem will be the key that unlocks many doors in calculus! 🌉📘

Comments

Post a Comment

If you have any queries, do not hesitate to reach out.
Unsure about something? Ask away—I’m here for you!

Popular posts from this blog

Understanding the Laplacian of 1/r and the Dirac Delta Function Mathematical Foundations & SageMath Insights

-
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...
Read more

Heuristic Computation and the Discovery of Mersenne Primes

-
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 \...
Read more

Neural Network Generalization in the Over-Parameterization Regime: Mechanisms, Benefits, and Limitations

-
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...
Read more