Maths: August 2023

Thursday, August 24, 2023

Discussions about the application of artificial intelligence in solving complex mathematical problems.

Absolutely, the topic of AI in Mathematics is quite intriguing and has been gaining attention due to its potential to revolutionize problem-solving and mathematical research. Here's some more information about this topic:

AI in Mathematics involves the use of artificial intelligence techniques, such as machine learning, deep learning, and automated reasoning, to tackle complex mathematical problems that might be computationally intensive, time-consuming, or even beyond human capabilities. This intersection has the potential to accelerate mathematical research, automate tedious calculations, and even discover new mathematical insights.

Here are some aspects that might be discussed under the hashtag #AIinMathematics:

Automated Theorem Proving: AI systems can be used to automatically prove mathematical theorems. These systems use logical reasoning and computational techniques to determine the validity of mathematical statements.

Symbolic Mathematics: AI can help manipulate and simplify complex symbolic expressions, making it easier to handle intricate equations and formulas.

Optimization Problems: AI techniques can be employed to solve optimization problems, which are prevalent in various fields, including engineering, economics, and science.

Data-Driven Insights: AI can analyze large datasets to identify patterns, trends, and relationships that might not be easily apparent through traditional mathematical approaches.

Exploring New Mathematical Concepts: AI algorithms can help explore mathematical spaces that are challenging for humans to navigate, potentially leading to the discovery of new conjectures and theorems.

Mathematical Modeling: AI can aid in creating more accurate and complex mathematical models for real-world phenomena, improving predictions and simulations.

Algorithm Design: AI can assist in designing algorithms for solving mathematical problems, optimizing their efficiency and accuracy.

Education and Learning: AI-powered educational platforms can provide personalized learning experiences, adapting to the individual's pace and level of understanding in mathematics.

Computer-Aided Mathematics: AI tools can assist mathematicians in performing calculations, verifying proofs, and exploring mathematical structures.

Collaboration and Communication: AI systems might facilitate collaboration among mathematicians by suggesting potential areas of research or connecting researchers with similar interests.

Remember, the field of AI in Mathematics is rapidly evolving, with new advancements and applications emerging regularly. Following this hashtag on social media platforms can help you stay updated on the latest breakthroughs, discussions, and insights in this exciting and interdisciplinary field.

7th Edition of International Conference on Mathematics and Optimization Method

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Complex Numbers and Basic Operations

Complex mathematics, often referred to as complex analysis, is a branch of mathematics that deals with complex numbers and their properties. Complex numbers are numbers that consist of both a real part and an imaginary part, and they are written in the form "a + bi," where "a" is the real part, "b" is the imaginary part, and "i" is the imaginary unit (defined as the square root of -1).

Complex analysis explores the properties and behaviors of functions that involve complex numbers. It encompasses a wide range of topics, including:

Complex Numbers: Understanding the properties of complex numbers, arithmetic operations (addition, subtraction, multiplication, division), and the representation of complex numbers in the complex plane.

Complex Functions: Studying functions of a complex variable, which map complex numbers to other complex numbers. Complex functions can be continuous, differentiable, and have properties like holomorphicity.

Complex Derivatives: Defining derivatives of complex functions and studying their properties, such as the Cauchy-Riemann equations that link the real and imaginary parts of the derivative.

Complex Integration: Investigating integration of complex functions along curves in the complex plane. Key concepts include contour integration and the Cauchy Integral Formula.

Residue Theory: Analyzing the behavior of functions using the residues of singularities (poles and essential singularities) in the complex plane. This is particularly useful in solving complex integrals.

Conformal Mapping: Exploring functions that preserve angles between curves, which have applications in fields such as fluid dynamics and electrostatics.

Complex Power Series: Representing functions as power series with complex coefficients, similar to Taylor series in real analysis.

Analytic Continuation: Extending the domain of a complex function to include points where it might not be initially defined, often leading to interesting and unexpected results.

Applications: Complex analysis has applications in various scientific and engineering fields, including physics, engineering, signal processing, fluid dynamics, and more.

Complex analysis is a fascinating and important area of mathematics that helps us understand and analyze functions that involve both real and imaginary components. It has profound applications in many scientific disciplines and is a fundamental part of advanced mathematics.

7th Edition of International Conference on Mathematics and Optimization Method

Tuesday, August 22, 2023

Making sense of the world through Maths

If you mime a silent Edvard Munch scream every time you see a mathematical problem, perhaps it is time to take a closer look at how much of our life is actually governed by numbers and equations. “Maths has the potential to connect us to universal truths that we can each access directly with our bodies and minds,” says Vijay Ravikumar, a mathematician with a penchant for puppetry and theatre arts.

Ravikumar recently premiered the “Geometry of Vision” series of online learning modules, which examines how we make visual sense of the world around us through Maths. The course has been developed with the support of the National Programme of Technology Enhanced Learning (NPTEL) and is available free online. In India, students can take it for credit at their home institution.
Maths to arts to maths again

Ravikumar’s own journey into interactive learning has an unusual background. Growing up in the U.S., he obtained a double major in Maths and English Literature from Amherst College, Massachusetts, before pursuing his Ph.D. in Maths at Rutgers University. He moved to India in 2013 for post-doctoral studies at the Tata Institute of Fundamental Research (TIFR) Mumbai, and then became an assistant professor at the Chennai Mathematical Institute.

By 2018, he left academia completely to become a theatre artist and puppeteer. However, Ravikumar shifted his focus back to Maths when the COVID-19 lockdown made online education de rigueur. “Geometry of Vision ”aims to demystify the way Maths works, especially in a non-academic setting. While we may not perceive ourselves as born number crunchers, our physical senses are constantly applying mathematical principles to process sensory data, says Ravikumar.

“Maths can be surprisingly intuitive at a visceral level, even if we’ve never thought about it consciously,” he says. His modules explain how the human mind is able to construct an intricate 3D understanding of our surroundings from 2D visual snapshots, by utilising various deep, elegant, and ancient mathematical principles. Beginning with an investigation of perspective drawing, and a survey of techniques artists have used for representing depth, the lessons move on to the “projective geometry” branch of Maths.
Open source learning

“My goal is to create a gateway to fundamental maths that is both accessible and exciting to interested learners from pretty much any background — whether they’re high-school or Ph.D. students, or adult learners looking for interesting activities after retirement,” says Ravikumar.

Keeping in mind that online instruction is a totally different medium from traditional in-person teaching, he has designed an interactive e-book format. “The course is not meant to be accessed through the usual NPTEL portal, but rather through an ‘Interactive Sandbox’ that I designed with Aatish Bhatia, a science writer at the New York Times. The student can scroll through a given lecture watching short, visually engaging videos interspersed with interactive questions, activities, and graphics. The ‘Interactive Sandbox’ is open source, and we plan to create a tutorial so interested teachers can easily adopt it,” he says.

Living and teaching in the U.S. and India has helped him appreciate the varied approaches towards education. “I hope this course can connect with a wide variety of learners, not only in terms of life experiences and mathematical backgrounds but also geographic locations. At the end of the day, one of the aspects I love most about Maths research is how it can bring together people of vastly different cultures, classes, and nationalities in a shared exploration,” he says.

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Monday, August 14, 2023

ChatGPT gets poorer in Maths, people say getting dumber with age is most humanlike thing it could do

A study conducted by Stanford University reveals that the popular AI chatbot ChatGPT created by OpenAI is experiencing fluctuations in performance, particularly in solving mathematical equations. The research highlights the phenomenon of "drift" in AI language models, emphasizing the importance of monitoring and addressing these changes to ensure consistent and reliable performance.

By India Today Tech: A study conducted by Stanford University found that the popular AI chatbot ChatGPT, created by OpenAI, is getting worse at solving mathematical equations. The chatbot experienced significant performance fluctuations on certain tasks between March and June. The research compared two versions of the technology– GPT-3.5 and GPT-4, focusing on tasks such as solving math problems, answering sensitive questions, generating software code, and visual reasoning.
As per the Fortune report, the study revealed a phenomenon called "drift," where the technology's ability to perform specific tasks changed unpredictably over time. In the case of GPT-4's math problem-solving capability, its accuracy dropped drastically from 97.6 per cent in March to a mere 2.4 per cent in June. As per the findings, the GPT-3.5 model demonstrated an opposite trajectory, improving from 7.4 per cent accuracy in March to 86.8 per cent in June on the same task.

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Thursday, August 10, 2023

Building India into a powerhouse in maths

Despite its rich legacy and size, India doesn’t fare well on global benchmarks of maths. Here’s a road map on how to do it

Mathematics is the foundation of modern science and technology, and its role in critical areas of national security such as cryptography is even more singular. However, despite a rich historical legacy (Aryabhatta, Brahmagupta and Bhaskara, for instance) as well as renowned mathematicians in the 20th century (such as S Ramanujan, SN Bose, PC reagents Mahanobis and CR Rao), India has fared less well in this critical field in recent years.

A recent study evaluated the contributions of top mathematicians based on the Discipline H-index or D-index, which tracks each scientist’s published scholarly papers and citations in their specific discipline.

Among the top 50 mathematicians, only one is of Indian origin; among the top 100, four are of Indian origin. Among the top 500, 15 are of Indian origin, but just one is based in India. Among the top 2,332 mathematicians, just 17 are based in India (less than 1%).

India ranks 19th, lower than tiny countries such as Israel, Austria and Belgium. While there have been two Indian-origin winners of the Fields medal (the Nobel Prize of mathematics) – Akshay Venkatesh and Manjul Bhargava – their parents had emigrated to Australia and Canada, respectively. And while institutions such as the Chennai Institute of Mathematics, Indian Institute of Science, Indian Statistical Institute and Tata Institute of Fundamental Research, have excellent mathematicians, the pool is thin.

Given India’s size, this is deeply dismaying. There are two reasons why India should be doing much better in mathematics. First, among all Science, Technology, Engineering and Mathematics (STEM) fields, the last one is the least capital-intensive. There is relatively less need for capital investments in labs and equipment, which need to be constantly upgraded. Mathematics also does not require large running costs of elaborate support infrastructure such as lab technicians or costly reagents.

Furthermore, if STEM is a ladder of social mobility, mathematics is even more so. Merit is rarely unambiguous, since the nature of standards and the referees that enforce them markedly shape its perception. But some fields of human endeavour have more unambiguous markers of merit.

In mathematics (along with sports, chess, and music), quality cannot be easily gamed. The language of maths is universal — and the standards of merit are unambiguous. The great Indian mathematician, Ramanujan, became a byword in number theory and pure mathematics despite his poverty, weak English language abilities, and being a fish out of water amidst the dons of Cambridge University.

India has failed to produce another Ramanujan despite the language of mathematics being universal (and, hence, English language proficiency mattering much less), a population that has grown five-fold, and is much more educated (at least as measured by the percentage of population finishing high school). This speaks volumes about India’s education system. Indeed, the current system of education is expressly designed to polish the stones and dim the diamonds.

Brilliant mathematicians are well likely to perform weakly or even fail in some subjects that don’t interest them. But India’s education prefers well-rounded mediocrity to narrow brilliance.

One of India’s biggest security challenges going forward will be cybersecurity and at the heart of it will be cryptography, for which the country needs hundreds of young talents in pure mathematics. Currently, key national security agencies such as the National Technical Research Organisation (NTRO) face serious human capital challenges (the largest employer of mathematicians in the US is the National Security Agency or NSA).

But the security challenge for India is much greater. Like all armies, the Indian armed forces are well prepared for the last war. This may matter less under conditions of slow technological change. But when technological change is rapid – and even more when it occurs at the breakneck speed as is the case today – the structures and personnel need a serious rethink.

To take an example, the US set up a new Army Futures Command (AFC) in 2018 (with about 20,000 personnel) to develop the technologies and concepts that will enable its armed forces to stay abreast of the sheer range and speed of unprecedented disruptive technologies that are impacting warfare. AFC is charged with leveraging developments in areas such as robotics, quantum computing, hypersonics, directed energy and Artificial Intelligence pioneered by the private sector. The key personnel are PhDs, who work, unlike the army’s usual hierarchical ethos. Knowledge and expertise, not rank and polish, matter. Breaking from the insularity that characterises militaries, it deliberately listens to a range of external civilian advisers, regularly convening meetings with technology experts.

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'Scissors congruence,' an ancient geometric idea that's still fueling cutting-edge mathematical research

In math class, you probably learned how to compute the area of lots of different shapes by memorizing algebraic formulas. Remember "base x height" for rectangles and "½ base x height" for triangles? Or "𝜋 x radius²" for circles?

But if you were in math class in ancient Greece, you might have learned something very different. Ancient Greek mathematicians, such as Euclid, thought of area as something geometric, not algebraic. Euclid's geometric perspective, recorded in his foundational work "Elements," has influenced research programs across centuries—even the work of mathematicians today, like the two of us.

Modern mathematicians refer to Euclid's concept of "having equal area" as "being scissors congruent." This idea, based on cutting up shapes and pasting them back together in different ways, has inspired interesting mathematics beyond just computing areas of triangles and squares. The story of scissors congruence demonstrates how classical problems in geometry can find new life in the strange world of abstract modern math.

Imagine you're back in math class and you have a pair of scissors, some tape and a piece of construction paper. Your teacher instructs you to make a new flat, two-dimensional shape using all of the construction paper and only straight-line cuts. Using your scissors, you cut the paper into a bunch of pieces. You start moving these pieces around—maybe you rotate them or flip them over—and you tape them back together to form a new shape.

Using your algebraic formulas for area, you could check that the area of your new shape is equal to the original area of the construction paper. No matter how a 2D shape is cut up—as long as all the pieces are taped back together without overlap—the area of the old and the new shape will always be equal.

For Euclid, area is the measurement that is preserved by this geometric "cutting-and-pasting." He would say that the new shape you made is "equal" to the original piece of construction paper—mathematicians today would say the two are "scissors congruent."

What can your new shape look like? Because you're only allowed to make straight-line cuts, it has to be a polygon, meaning none of the sides can be curved.

Could you have made any possible polygon with the same area as your original piece of paper? The answer, amazingly, is yes—there's even a step-by-step guide from the 1800s that tells you exactly how to do it.

In other words, for polygons, Euclid's notion of area is exactly the same as the modern one. In fact, you may have even used Euclid's idea of area before in computations without knowing it.

For example, you can use scissors congruence to compute the area of a pentagon. Since area is preserved if you cut the pentagon up into smaller triangles, you can instead find the area of these triangles (using "½ base x height") and add them up to get the answer.

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Wednesday, August 9, 2023

Demand Forecasting Methods for Inventory Optimization.

Inventory management is a critical aspect of business operations that involves tracking and controlling a company's inventory levels to ensure efficient operations and minimize costs. Here are some statistical concepts and techniques commonly used in inventory management:

Demand Forecasting:Moving Averages: A simple technique that calculates the average demand over a specific time period to smooth out fluctuations and identify trends.
Exponential Smoothing: A method that assigns exponentially decreasing weights to past data points, giving more weight to recent data for better responsiveness to changes.

Reorder Point (ROP):Safety Stock: Safety stock is an extra amount of inventory held to mitigate the risk of stockouts due to unexpected variations in demand or lead time.

Economic Order Quantity (EOQ):This is the optimal order quantity that minimizes the total cost of inventory, considering both ordering costs and carrying costs.

ABC Analysis:This categorizes inventory items into three groups (A, B, and C) based on their value and contribution to overall inventory costs. This helps prioritize management efforts.

Lead Time Analysis:Statistical analysis of the time it takes from placing an order to receiving the inventory. This helps in setting reorder points and safety stock levels.

Inventory Turnover Ratio:This ratio indicates how many times a company's inventory is sold and replaced over a specific period. A higher turnover ratio often indicates more efficient inventory management.

Stockout Rate:This metric calculates the frequency or probability of running out of stock for a particular item.

Service Level:The desired probability of not having a stockout, often denoted as a percentage. It helps determine the appropriate level of safety stock.

Statistical Quality Control:Techniques like control charts help monitor the quality of incoming inventory and identify any deviations from the expected norms.

Demand Variability Analysis:Statistical measures such as standard deviation or coefficient of variation are used to quantify the variability in demand, which affects safety stock calculations.

These are just a few examples of how statistics are used in inventory management. The goal is to optimize the balance between holding costs and stockout costs while ensuring products are available when needed. Different businesses might use various techniques based on their industry, products, and specific requirements.

7th Edition of International Conference on Mathematics and Optimization Method