Gonit Sora

Analysis of Random Forests in predicting the number of 6s hit in IPL

Satvik Dutt — Fri, 22 Nov 2024 06:31:11 GMT

INTRODUCTION

Cricket, with its global popularity and complex gameplay, is very fertile ground for statistical analysis. The many quantitative elements offer unique opportunities for analytical observations. The use of statistical analysis in cricket has become crucial, whether it is informing team predictions, player selection, match outcomes, or even fan predictions (Cricket Webs, 2023). As the demand for deeper insights grows, traditional statistical methods are being aided by machine learning algorithms that can identify patterns and make data-driven predictions.

Despite their growing popularity in other areas, random forests, an ensemble learning approach (Breiman,2001), haven’t gotten as much attention in cricket analytics compared to other machine learning and artificial intelligence techniques. This means there's a lot of potential to be unlocked using Breiman's original algorithm and its variations to analyze cricket data, especially in shorter format games with multiple rounds and datasets.

The Indian Premier League (IPL) is a T20 cricket league that has gained massive popularity throughout the world. It displays some of the best cricket players worldwide and offers a wealth of statistical data that can be used for detailed analysis. The dataset that was used in this research is made up of carefully recorded performance and match outcomes in addition to all sorts of in-game statistics that have been gathered through 4 IPL seasons from the years 2018 to 2022.

This article examines different machine learning algorithms in cricket and explores the effect of different powerful algorithms within cricket data. While several algorithms offer valuable tools for such analysis, we will take a look at how random forests emerge as a very accurate predictor of the number of sixes hit by an individual.

RANDOM FORESTS

Random forests have emerged as a powerful ensemble learning technique within the field of machine learning. It is considered an ensemble learning algorithm because it combines the predictions from multiple machine learning algorithms to make more accurate predictions than any individual model. Specifically, a Random Forest builds many decision trees and merges them to get a more accurate and stable prediction.

Random forests build on the idea of bagging. This involves creating multiple classification and regression trees using random subsets of the training data (with replacement). This approach combats the instability of individual classification and regression trees, which can be overly sensitive to small changes in the data (Ziegler and König, 2011).

ANALYSIS OF DATA IN CRICKET

The Area Under the Curve (AUC) is among the key metrics of machine learning, regardless of the primary focus of the evaluation of the classification model. It is derived from the Receiver Operating Characteristic (ROC) curve, which is a graphical representation of a model's performance across various threshold settings. Accuracy, too, is a very familiar term in the context of machine learning, especially when working with classification problems. Accuracy is a fraction showing how many predictions are correct out of all the predictions being made.

1 . Boxplot of Accuracy for 20 train/test splits for predicting if a player hit over five sixes in the season.

2. Boxplot of AUC values.

The dataset was pre-processed and split into training and test sets. The Random Forest, SVM, and XGBoost models were then trained on the training data, followed by predictions on the test data. . Accuracy was plotted to assess the correct classification of sixes, while AUC was used to evaluate the models' classification capability across varying decision thresholds.

Random Forests is shown to be superior compared to other models like Support Vector Machines (Cortes and Vapnik, 1995), BART (Chipman et al., 2010), and XGBoost (Chen and Guestrin, 2016). The boxplot has been used to compare accuracy and AUC (Area Under the Curve) scores. The accuracy boxplot in Figure 1 suggests that Random Forests are more powerful and have higher median accuracy values (0.92-0.98) than the rest in predicting a binary outcome of whether a player hits over five sixes in the season, making it a superior performance and stability model. In the same manner, the AUC boxplot in Figure 2 exhibits the ability of Random Forests to apply more accurate classifications, showing a more compact distribution with the AUC median values (0.97-0.99) than SVM and XGBoost. These results suggest that Random Forests are more effective in capturing the nuances of the dataset, likely due to their ensemble learning approach, which reduces overfitting and enhances predictive performance.

CONCLUSION

Preliminary findings indicate that RF is more effective in calculating if a player hit over five sixes in the season or not in comparison to other algorithms. Its capability to work with different kinds of data, like numerical and categorical data with IPL characteristics, makes it versatile for adding up different data. The inherent randomness in tree construction helps in avoiding overfitting, a common challenge in predictive modeling.

By aggregating the predictions of multiple trees, Random Forests achieve superior predictive accuracy. This ensemble approach reduces the impact of outliers in the data, leading to more reliable estimates of sixes. Having a reliable estimate of sixes not only enhances team strategies and player evaluations but also empowers coaches and broadcasters to efficiently make informed decisions from player retention to marketing campaigns.

REFERENCES

Breiman, L. (2001). Random Forests. Machine Learning, 45(1):5–32.

Chipman, H. A., George, E. I., and McCulloch, R. E. (2010). Bart: Bayesian additive regression trees. The Annals of Applied Statistics, 4(1):266–298.

Cortes, C. and Vapnik, V. (1995). Support-vector networks. Machine learning, 20(3):273–297

Chen, T. and Guestrin, C. (2016). Xgboost: A scalable tree-boosting system. In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining, pages 785–794. ACM

Ziegler, A. and K¨onig, I. R. (2011). Mining data with random forests: current options for real-world applications. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 1(1):55–63

Cricket Webs. (2023). Cricket Technology: How Data and Analytics Are Shaping the Sport.

Analyzing the Brachistochrone Problem for the Conic Sections and its real life applications

Aarush Patel — Tue, 22 Oct 2024 07:02:38 GMT

Analyzing the Brachistochrone Problem for the Conic Sections and its real lfie applications

Aarush Patel

August 2024

Abstract

This paper explores the brachistochrone problem for several curves. Throughout the paper, we find the time taken by an object of any mass to travel from the initial position $(1, 0)$ to the final position $(0, 1)$ through any curve. Finding a generalized integral time formula applicable to all curves of the form $y = f (x)$ , we further apply the formula to a line and the following conic sections - parabola of form $y = a x^{2} + b x + c$ and $x = a y^{2} + b y + c$ , , circle of the form $(x - a)^{2} + (y - b) = r^{2}$ , ellipse with $x$ and $y$ as major axes, and hyperbola with $x$ and $y$ as major axis. The parabola with equation $x = a y^{2} + b y + c$ was found to be the curve of fastest descent among all the conic sections. These findings are also observable in real life scenarios. For example, during descent, roller coasters are generally parabolic as they would provide more thrill to the visitors than a straight one.

1 Introduction ..... 1
2 Generalized integral time formula for any curve and descent for a straight line ..... 2
3 Time taken for the descent by various conic sections ..... 3
3.1 Parabola ..... 4
3.2 Circle ..... 4
3.3 Ellipse ..... 5
3.4 Hyperbola ..... 7
4 Conclusion ..... 8

1 Introduction

The brachistochrone problem, proposed by Johann Bernoulli in 1696, aims to find, for points

A

and

B

with

A

located above

B

, a curve such that descent of an object due to gravity along that curve takes the shortest possible time. The problem considers only gravitational force on the object and ignores other forces such as friction. It was solved by a number of famous mathematicians such as Johann Bernoulli himself, Jakob Bernoulli, and Isaac Newton, all of them proposing a different approach [4]. Till date, numerous researchers have tried to expand upon the problem by including external factors such as non-uniform gravitational field [1], friction [2] and some other challenges, examining the problem further. These methods have been quite effective but require heavy machinery such as the calculus of variations and the Euler-Lagrange differential equation. Additionally, all of the papers are predominantly focused on deriving the equation of the curve with quickest descent and are not focused on other curves. In this paper, we focus on a more elementary approach and study descent along the conic sections from a point

A

with coordinates

(0, 1)

to a point

B

with coordinates

(1, 0)

. In Section 2, we utilise a general approach for calculating the time it takes to descend along a curve defined by a function

y = f (x)

. In the following section, 3 we give a detailed account of the time period in cases of parabola, circle, ellipse, and hyperbola respectively. Using an online tool [5] to find the value of integrals, we compare the findings of time periods of different curves. The curves when arranged in decreasing order of their time periods are as follows : line > parabola of form

y = f (x) =

hyperbola with

y

axis as major axis

>

ellipse with

y

axis as major axis

>

circle

>

ellipse with

x

axis as major axis

=

hyperbola
with

x

axis as major axis

>

parabola of form

x = f (y)

. Applying these findings in real life, we find that water parks generally have curved slides to provide a faster and more enjoyable experience to the person. Additionally, we understand that the shortest curve is not always the fastest one.
The methodology mainly comprises of independent calculations for calculating the time. Additionally, a small literature review was done to gather historical background on the problem.

2 Generalized integral time formula for any curve and descent for a straight line

Let us take an object of mass

m

sliding from point

A

to point

B

. Throughout the paper, we will consider the initial position position

(A)

to be

(0, 1)

and the final position

(B)

to be

(1, 0)

. In order to find the time taken, we will first use the law of conservation of energy. The energy at the initial point is just the gravitational potential energy

m g h

where

h

is the height. We have

h = 1

since our initial point is

(1, 0)

. The energy at any other point

P

on the curve will be

\frac{1}{2} m v^{2} + m g y

where

v

is the velocity of the object at

P

and

h

is the

y

-coordinate. By conservation of energy,

\begin{matrix} m g * 1 = \frac{1}{2} m v^{2} + m g y \\ \frac{1}{2} m v^{2} = m g (1 - y) \\ v = \sqrt{2 g (1 - y)} \end{matrix}

We have found the velocity at any point. By definition, velocity is the derivative of displacement with respect to time i.e

v = \frac{d s}{d t}

. Rearranging gives

d t = \frac{d s}{v}

and integrating we obtain

t = \int_{0}^{1} d t = \int_{0}^{1} \frac{d s}{v} = \int_{0}^{1} \frac{d s}{\sqrt{2 g (1 - y)}}

Note that

d s

is the small change in displacement in a small period of time

d t

. Our aim is to express

d s

in terms of

d x

. In the below graph, the blue line represents the curve

y = x^{2}

and the purple line refers to the tangent to the curve at

x = 1

. The angle made by the tangent line with the positive

x

axis is denoted by

θ

. Hence, the slope of the tangent line is equal to

\tan θ

. Through the right angled triangle formed with

d s

as the hypotenuse and

d x

as the base,

d s = \frac{d x}{\cos θ}

Now, the derivative of the curve at

x = 1

is the slope of the line i.e.

\tan θ

. Now from equation

\sec^{2} θ =

\tan^{2} θ + 1

we have

\cos^{2} θ = \frac{1}{\tan^{2} θ + 1}

θ \leq \frac{π}{2}, \cos θ

is positive. Now

\cos θ = \frac{1}{\sqrt{\tan^{2} θ + 1}} = \frac{1}{\sqrt{{(\frac{d y}{d x})}^{2} + 1}}

So,

d s = \sqrt{({(\frac{d y}{d x})}^{2} + 1)} d x

and

\begin{matrix} (2.1) & t = \int_{0}^{1} \frac{\sqrt{1 + {(\frac{d y}{d x})}^{2}}}{\sqrt{2 g (1 - y)}} d x \end{matrix}

This is the general time formula, which is applicable to any curve

y = f (x)

.
Consider the case of a line passing through the points

(1, 0)

and

(0, 1)

. It is given by

y = 1 - x

so that

\frac{d y}{d x} = - 1

. Hence, the time taken by an object to roll down the line

y = x - 1

from

(1, 0)

(0, 1)

is equal to

\int_{0}^{1} \sqrt{\frac{1 + (- 1)^{2}}{2 g (1 - 1 + x)}} d x = \int_{0}^{1} \sqrt{\frac{2}{2 g x}} d x = {\frac{2 \sqrt{x}}{\sqrt{g}} |}_{0}^{1} = \frac{2}{\sqrt{g}} = 0.638877 seconds

(Taking the value of g as 9.8 ).

Figure 2.1: Graph of

y = x^{2}

along with its tangent

y = 2 x - 1

x = 1

3 Time taken for the descent by various conic sections

In this section, we discuss the time taken by an object of any mass to travel from

(1, 0)

(0, 1)

through various conic sections listed below.

Figure 3.1: A graph summarising the curves, which are labelled in the order of increasing descent time.

The code for the above graph is accessible at https://github.com/Aarush2012/Brachistochrone-curvegraph.git.

3.1 Parabola

Consider a parabola of the form

y = a x^{2} + b x + c

passing through the points

(1, 0)

and

(0, 1)

. Let us substitute the coordinates and find the equation of such a parabola. We get:

{\begin{cases} a (1)^{2} + b (1) + 1 = 0 \\ c = 1 \end{cases}

The general equation for a parabola passing these points is

y = a x^{2} - (a + 1) x + 1

We can now apply the general formula (2.1). First, we differentiate:

\frac{d y}{d x} = 2 a x - a - 1

. Substituting, we get

t (a) = \int_{0}^{1} \sqrt{\frac{1 + (2 a x - a - 1)^{2}}{2 g (1 - a x^{2} + (a - 1) x - 1)}} d x

Note that the time

t

is dependent on the value of the parameter

a

, and we can try to find the minimum of

t (a)

to minimize the descent time.

We calculate the value of the above integral by plotting it into Desmos [5]. As

a

is a variable, we add a slider to change the value of

a

. The observations are that the value of the integral is the least when

a = 1

with time taken for descent equal to

0.595227 \dots

seconds. At

a = 0.9

, the value comes out be 0.5958 and at 1.1 , it is 0.59543 . From

a = 0

a = 1

, the time taken decreases and after 1 , it keeps on increasing. The curve is represented by

y = x^{2} - 2 x + 1 = (x - 1)^{2}

Now, let us discuss the time taken by a parabola of the form

x = a y^{2} + b y + c

. Substituting the coordinates in the equation,

c = 1

and

a + b + 1 = 0

so that

b = - 1 - a

. The parabola is of the form

x = a y^{2} - (a + 1) y + 1

. Differentiating the equation with respect to

y, \frac{d x}{d y} = 2 a y - a - 1

. Our second step is to express

y

in terms of

x

. For this, we will express

x

in terms of a perfectly squared equation, which will help us isolate

y

. We have

x = a y^{2} - (a + 1) y + \frac{(a + 1)^{2}}{4 a} + 1 - \frac{(a + 1)^{2}}{4 a} = {(\sqrt{a} y - \frac{a + 1}{2 \sqrt{a}})}^{2} + \frac{4 a - (a + 1)^{2}}{4 a}

. Therefore

\pm \sqrt{\frac{4 a x - 4 a + (a + 1)^{2}}{4 a}} = \sqrt{a} y - \frac{a + 1}{2 \sqrt{a}}

so that

y = \frac{\pm \sqrt{\frac{4 a x - 4 a + (a + 1)^{2}}{4 a}} + \frac{a + 1}{2 \sqrt{a}}}{\sqrt{a}}

Clearly, the graph of the descent curve cannot pass through the second quadrant. In addition, it has to pass through the points

(0, 1)

and

(1, 0)

. This implies that in the equation above we need to choose the minus sign and restict

a

to the interval

a \in [0, 1]

. So we have

y = \frac{- \sqrt{\frac{4 a x - 4 a + (a + 1)^{2}}{4 a}} + \frac{a + 1}{2 \sqrt{a}}}{\sqrt{a}} = \frac{a + 1 - \sqrt{4 a x - 4 a + (a + 1)^{2}}}{2 a}

Now, we substitute the values of

\frac{d y}{d x}

and

y

in the equation (2.1), we obtain an integral depending on

a

. The least value of this integral is

0.58377 \dots

. which occurs at

a = 0.91

3.2 Circle

Consider a circle of

(x - a)^{2} + (y - b)^{2} = r^{2}

passing through the points

(1, 0)

and

(0, 1)

. Here,

a

stands for the

x

-coordinate of the centre and

b

refers to the

y

coordinate of the centre;

r

is the radius of the circle. Substituting the points in the equation we get

{\begin{cases} (1 - a)^{2} + b^{2} = r^{2} \\ a^{2} + (1 - b)^{2} = r^{2} \end{cases}

Equating the left sides, we have

1 - 2 a + a^{2} + b^{2} = a^{2} + 1 - 2 b + b^{2}

so that

a = b

. Now the first equation gives

r^{2} = 2 a^{2} - 2 a + 1

and we have

(x - a)^{2} + (y - a)^{2} = 2 a^{2} - 2 a + 1

We can rewrite it as follows:

x^{2} + y^{2} - 2 a (x + y - 1) - 1 = 0

Differentiating with respect to

x

, we get

2 x + 2 y \frac{d y}{d x} - 2 a (1 + \frac{d y}{d x}) = 0

so that

\frac{d y}{d x} = \frac{a - x}{y - a}

In the integral, all the

y

terms should be represented in terms of

x

so that we can integrate with respect to

x

. We express

y

y = - \sqrt{2 a^{2} - 2 a + 1 - (x - a)^{2}} + a

We use the negative sign because it represents the lower part of the circle, which is used to traverse from

(1, 0)

(0, 1)

. As for the parabola case, we substitute

\frac{d y}{d x}

and

y

in equation (2.1). The least value of the integral is

0.58512 \dots

which appears when

a = 1.3

. An important point to consider here is that

a \geq 1

. This is due to the fact that when

a < 1

the circle no longer uses its lower part to pass through those points. Instead, the graph will be represented by

{y = \sqrt{2 a^{2} - 2 a + 1 - (x - a)^{2}} + a

, which is the upper part of the circle. Figure 3.2 graphically demonstrates the above idea.

Figure 3.2: Curves represented graphically demonstrating why the negative sign was chosen for positive values of

a

Now, for the case

a < 1

, we find the value of the integral by using the corresponding

y

value 3.2 . The integral's value keeps on decreasing as x becomes more negative until it reaches a limiting value of

0.6388 \dots

seconds. This is same as the least time taken by a line as when

x \to - \infty

, the part of the circle joining

(1, 0)

and

(0, 1)

becomes increasingly flatter, eventually taking the shape of a line.

3.3 Ellipse

Consider an ellipse with x -axis as its major axis and

x, y

coordinate of its center as

x_{0}

, 1 . i.e.

\frac{{(x - x_{0})}^{2}}{a^{2}} + \frac{(y - 1)^{2}}{b^{2}} = 1

passing through the points

(1, 0)

and

(0, 1)

. Substituting the coordinates in the equation, we get

{\begin{cases} \frac{{(1 - x_{0})}^{2}}{a^{2}} + \frac{1}{b^{2}} \\ \frac{x_{0}^{2}}{a^{2}} + \frac{0}{b^{2}} = 1 \to x_{0}^{2} = a^{2} \end{cases}

. Inserting the value of

x_{0}

+ a

in the first equation,

\frac{(1 - a)^{2}}{a^{2}} + \frac{1}{b^{2}} = 1

So,

\frac{1}{b^{2}} = 1 - \frac{(1 - a)^{2}}{a^{2}} = \frac{- 1 + 2 a}{a^{2}}

∴ b^{2} = \frac{a^{2}}{2 a - 1}

. So , the equation of the ellipse in this case is

\begin{matrix} (3.1) & (x - a)^{2} + (y - 1)^{2} (2 a - 1) = a^{2} \end{matrix}

We take the value of

x

+ a

as the ellipse is defined only for positive

a

values. . Differentiating the above equation with respect to

x, 2 (x - a) + 2 (2 a - 1) (y - 1) \frac{d y}{d x} = 0

\frac{d y}{d x} = \frac{a - x}{(2 a - 1) (y - 1)}

. The next step will be to isolate y.

(y - 1)^{2} = \frac{a^{2} - (x - a)^{2}}{2 a - 1}

\begin{matrix} (3.2) & y = - \sqrt{\frac{2 a x - x^{2}}{2 a - 1}} + 1 \end{matrix}

. Due to the same reason in the curves discussed, we choose the negative sign. Plugging the respective values of

\frac{d y}{d x}

and

y

into the time period equation 2.1 , we find that value of this integral varies inconsistently for values of

a

. However, gradually, with a huge increase in the value of a, there is a decreasing trend in the value of this integral. Hence, as

a \to \infty

, the value of the integral reaches a limiting value. Let's go back to the original equation we derived 3.1. Expanding it, we obtain

x^{2} - 2 a x + a^{2} + (2 a - 1) (y - 1)^{2} = a^{2}

. As our aim is to examine the curve near

x = 0

and

x = 1, x^{2}

can be neglected.

∴ 2 a x = (2 a - 1) (y - 1)^{2}

. Now,

2 a = 2 a - 1

as we are finding the equation in which

a \to \infty

\begin{matrix} (3.3) & x = (y - 1)^{2} \end{matrix}

We expressed an ellipse of type 3.1 with a significantly large value of

a

in the form of a parabola. Expressing the equation 3.3 in terms of

x

\begin{matrix} (3.4) & y = \pm \sqrt{x} + 1 \to y = - \sqrt{x} + 1 \end{matrix}

\cdot \frac{d y}{d x} = - \frac{1}{2 \sqrt{x}}

. Hence, the least time taken by an object to reach from

(1, 0)

(0, 1)

through

3.3 =

\int_{0}^{1} \sqrt{\frac{1 + \frac{1}{4 x}}{2 * 9.8 (1 + \sqrt{x} - 1)}} d x = 0.58436 \dots

. seconds.
Let us take the case of an ellipse with y axis as its major axis and center as

(1, y_{0})

. The equation of such an ellipse would be

\frac{{(y - y_{0})}^{2}}{a^{2}} + \frac{(x - 1)^{2}}{b^{2}} = 1

. Applying the former strategy,

{\begin{cases} \frac{y_{0}^{2}}{a^{2}} = 1 \to y_{0}^{2} = a^{2} \\ \frac{(1 - a)^{2}}{a^{2}} + \frac{1}{b^{2}} = 1 \to b^{2} = \frac{a^{2}}{2 a - 1} \end{cases}

. Plugging these values in the equation,

\frac{(y - a)^{2}}{a^{2}} + \frac{(x - 1)^{2} (2 a - 1)}{a^{2}} = 1

Hence,

{(y - a)^{2} + (2 a - 1) (x - 1)^{2} = a^{2}

.
Differentiating with respect to

x, 2 (y - a) \frac{d y}{d x} + 2 (2 a - 1) (x - 1) = 0

\frac{d y}{d x} = \frac{(1 - 2 a) (x - 1)}{(y - a)}

For all,

a \geq 1

\begin{matrix} (3.5) & y = - \sqrt{a^{2} - (x - 1)^{2} (2 a - 1)} + a \end{matrix}

. Using the equation 2.1 T to find the time formula, we find that the least time is observed to occur at

a = 1.375

which is

0.588349 \dots

seconds. For

a > 2.2

, the value of the integral is seen to have values slightly more than 0.59 , and for

a \in [1, 2.2]

, the value decreases until

a = 1.375

and then starts to increase.

Figure 3.3: Graph of ellipse of type 3.1 with

a = 100

and parabola

x = (y - 1)^{2}

Figure 3.4: Graph of ellipse of type (3.1) with a

= 500

and parabola

x = (y - 1)^{2}

3.4 Hyperbola

In the previous section, while deriving the equation of the ellipse 3.1 , we equated

x

+ a

because when

a

becomes negative, the ellipse takes the form of a hyperbola. Hence, for a hyperbola with x axis as its major axis and center as

(x_{0}, 1)

, the time taken by an object to pass through it can be defined as 3.3 with negative values of

a

Figure 3.5: Graph of equation 3.1 with positive value of

a

Figure 3.6: Graph of equation 3.1 with negative value of

a

This integral also has a gradually decreasing trend with a large decrease in the value of

a

. Hence , as

a \to - \infty

, the integral approaches a limiting value. In the equation

3.2, x^{2}

can be neglected as it is drastically smaller than

2 a x

. Also,

2 a

can be equated to

2 a - 1

a \to - \infty

. Hence, the equation becomes

y = 1 - \sqrt{x}

This is same as the equation we derived for the ellipse 3.4. Hence, the least time will be same as that of the ellipse with

x

axis as major axis i.e. 0.58436 seconds.

Now, for an object to pass through a hyperbola with y axis as major axis and centre at

(1, y_{0})

from

(1, 0)

(0, 1)

, we will again choose the negative values of

a

in the equation for the ellipse 3.3 with y axis as major axis. However, the same curve used for the ellipse won't be applicable to the hyperbola. This is due to the fact that the curve 3.5 which passes through

(1, 0)

and

(0, 1)

does not through pass through those points for negative values of

a

Figure 3.7: Graph of equation3.5 for positive values of

a

Figure 3.8: Graph of equation 3.5
for negative values of

a

Instead, the upper part i.e.

\begin{matrix} (3.6) & y = \sqrt{a^{2} - (x - 1)^{2} (2 a - 1)} + a \end{matrix}

will be the equation for the hyperbola. Finding the value of the time integral 2.1 by plugging the values of

\frac{d y}{d x}

and

y

, we find that for all negative values of

a

, we find that there is also a decreasing trend in the its value. As,

a

approaches

- \infty

, the integral reaches a limiting value. Rearranging equation 3.6,

(y - a)^{2} = a^{2} - (x - 1)^{2} (2 a - 1) \to y^{2} - 2 a y + a^{2} = a^{2} - (x - 1)^{2} (2 a - 1)

. We want to work with the curve near the points

(0, 1)

and

(1, 0)

.Hence,

y^{2}

can be considered negligible.

∴ 2 a y = (x - 1)^{2} (2 a - 1) .2 a = 2 a - 1

a \to - \infty

. Hence, the curve will be

y = (x - 1)^{2} \cdot \frac{d y}{d x} = 2 (x - 1)

. So, the time integral

=

\int_{0}^{1} \sqrt{\frac{1 + 4 (x - 1)^{2}}{2 * 9.8 (1 - (x - 1)^{2})}} d x

The least time taken by an object of any mass to travel from the point

(1, 0)

(0, 1)

through hyperbola 3.6 ,

= 0.595227

seconds.

4 Conclusion

Comparing the time taken by an object to pass through a line and various conic sections, we observe that a parabola of the form

x = f (y)

is the curve with the quickest descent. However, we have restricted the number of parameters in the cases of ellipse and hyperbola by assuming them to be centered at

(x_{0}, 1)

and

(1, y_{0})

for those with

x

and

y

axes as major axis respectively. This problem can be applied in fields such as architecture and engineering, so as to design a quicker route. Further studies could be done to validate this paper's findings and expand upon the research by finding the least time to traverse any ellipse and hyperbola. This Additionally, using the parametric equations of a general cycloid, a similar method can be employed to prove why the cycloid is the curve of quickest descent.

References

[1] Mohammed Lufti, H. E. "The Effect of Gravitational Field on Brachistochrone Problem " . Phys: Conf. Ser. 1028012060
[2] Kurilin, Alexander"Brachistochrone and Sliding with Friction". Reseach Gate : 10.48550/arXiv. 2305.18345
[3] Esam H. Abdul-Hafidh "A new approach to solve the Brachistochrone problem by constructing a lattice unit cell" Heliyon, Volume 8, Issue 12, 2022, e11994, ISSN 2405-8440
[4] "Brachistochrone curve", Wikipedia, 23 June 2024
[5] Desmos, Amherst, 2011

The Fourier Transform: Bridging Theory and Applications in Signal Processing, Music Synthesis, and Climate Analytics

Arhaan Goyal — Thu, 26 Sep 2024 11:10:32 GMT

The author Arhaan Goyal is a student of Jayshree Periwal International School, Class of 2025, and can be reached at arhaangoyal[at]gmail.com.

This paper below explores the fundamental concepts and applications of the Fourier Transform, a convolution tool in mathematics, physics, and engineering. By establishing the connection between the Taylor series, Fourier series, and the Fourier Transform, we provide a solid foundation for understanding the transform's versatility. The paper highlights the expansion of the Fourier Transform, which is analogous to the Taylor series expansion, and utilizes orthogonality properties of the Fourier series representation. The paper also explores the applications of the Fourier Transform in signal processing, music synthesizing, and climate analytics.

An Incredibly Short Tale About Human Imagination

Gonit Sora — Mon, 13 May 2024 08:34:41 GMT

By Dr Mridu Prabal Goswami (Indian Statistical Institute, North-East Centre, Tezpur)

Author's Note: The views expressed in this piece are personal. The author
is extremely thankful to Mousumi Kalita Sachdeva and Abinash Panda.

Author's Preface: This piece is only a snapshot of the evolution and the articulation of human imagination that uses the language of Mathematics as the medium of expression. Being an outsider, i.e., not having any formal degree in Mathematics, is a serious limitation of the author. The emotional conflict that ensues from this limitation about whether to write this piece remains unresolved. Naturally, this piece does not intend to teach Mathematics. This piece only carries a personal wish that developers of this language feel what they do is so incredibly beautiful that even an outsider can appreciate it.

Yuval Noah Harari, the author of 'Sapiens A Brief History of Humankind', says language has made homo sapiens what it is now (see Araki. N. (2019): Sapiens and Language, Bull. Hiroshima Inst. Tech. Research, 53, 1-10). The tool of language has enabled Homines sapientes to express the belief that Homines sapientes are a space-faring species. A tiger however strong physically it is, may not have the language to express the wish "tigers were a space-faring species'' (The author does not intend to disrespect tigers and or any other animal species. They are mentioned only to bring out the juxtaposition of two species). Homines sapientes have formed, articulated, and long started their efforts to actualize such wishes. The language of Mathematics is the language that will be used to design ships that may enable the homines sapientes to visit far-off places. Two of the most important rules of this language are the rules of calculus and numbers. The following caveat is noteworthy: tigers may have the ability to hide technology from humans. However, an offended tiger may roar in displeasure and assert that tigers also use calculus and numbers to optimize the numbers and distances of possible prey from their locations during their hunt. While this maybe true, homines sapientes have written down the rules of calculus and numbers, which tigers are welcome to verify if they can and have the time to do so. If the tigers have also written down their calculus, then maybe it is advisable to make their research open access so that other species can also access the benefits. Since the author is not familiar with any journal, book, or lecture video on Mathematics by the tigers or any other species, the author shall limit their attention to Mathematics by homines sapientes. However, the author shall make occasional references to the tigers so that the conversation with another species is not lost.

It maybe feasible to start a discussion of the history of human imagination when the wheel was invented. The circular geometry of this object might have helped a generation of human ancestors to transport goods in their carts quite easily compared to the generations before who did not have access to the benefits of this circular geometry. Due to the incapability of the author to go that far back in time and talk about human imagination, this tale starts with the invention of what is currently known as modern calculus (The author does not have any opinion about whether Mathematics is discovered or invented. The choice of the word `invented' is the outcome of a random experiment.)

Imagine a function $f:{1,2}\rightarrow \mathbb{Q}$ defined by $f(1)=5,f(2)=10$, where $\mathbb{Q}$ denotes the set of rational numbers. Further, let ${1,2}$ be not some arbitrary set of symbols, and instead
${1,2}\subseteq \mathbb{Q}$. Let us ask an undefined question, "Is $f$ continuous?''. We can change the status of the question from undefined to defined by defining a distance metric on $\mathbb{Q}$. Let the distance between $x,y \in \mathbb{Q}$, be $d(x,y)=|x-y|$. Now we can apply the standard definition of continuity of a function for metric spaces that uses convergent sequences.
Thus, we need to check if $x_{n}\rightarrow x$, where $x_n\in {1,2}$ for all $n$, and $x\in {1,2}$, then $f(x_{n})\rightarrow f(x)$, where $f(x_{n})\in \mathbb{Q}$ for all $n$, and $f(x)\in \mathbb{Q}$. Since the only sequences $\{x_{n}\}_{n=1}^{\infty}$ that converge in ${1,2}$ are the sequences for which either $x_n=1$ after some $n$ or $x_n=2$ after some $n$, the only sequences that are relevant in the range of the function are those with $f(x_n)=5$ after some $n$ or $f(x_n)=10$ after some $n$. Since sequences that are eventually constant converge, the function $f$ is continuous.
However, the function does not "look'' like a function that has no breakpoints.

Let us do a thought experiment. Let us imagine that one of the homines sapientes a long time ago is sitting on the bank of a river, this homo sapiens who has nothing much to do picks up a pebble and draws a curve on the sand, has an epiphany about curves with no breakpoints, returns home with the exuberance of writing down Mathematics of no breakpoints, also passes a smirk at the resting tiger while returning home, sits down to write down the Mathematics, but then realizes that a notion of the continuum is not at its disposal yet. The snooty sapiens who had passed a smirk at the resting tiger realizes the importance of the continuum. If the imaginary snooty sapiens could invent distance metrics and continuous functions without inventing the continuum, it could be considered remarkable. Possibly this did not happen, and the homo sapiens might have received a smirk back from the resting tiger. However, the news that maybe conveyed to the resting tigers is that homines sapientes did invent the continuum called the set of real numbers denoted by $\mathbb{R}$. Under the distance function $d$ defined above, $\{1,2\}$ and $\mathbb{Q}$ are subspaces, i.e., special cases of $\mathbb{R}$ with distance function $d$. But hang on, another curious sapiens asks, if continuity of a function can be defined without using the continuum, then can integration and differentiability also be defined without using the continuum.

Consider the function $f$ again. Let us assign weights to $1$ and $2$. Let the weights be $\omega(1)=\frac{1}{3}$, $\omega(2)=\frac{2}{3}$ respectively. Then define $\int_{{1,2}}f(x)d\omega(x)=f(1)\omega(1)+f(2)\omega(2)=1\frac{1}{3}+2\frac{2}{3}=\frac{5}{3}$. However, making sense of a differentiable function in the standard sense is difficult without the continuum. Naively, let for $a\in \{1,2\}$ the derivative of $f$ at $a$ be $f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$. This seems outrageous. Since the only sequence that converges in $\{1,2\}$ are those sequences that are constant after some $n$, the ratio $\frac{f(x)-f(a)}{x-a}=\frac{0}{0}$ after some $n$ and thus undefined. To make sure that $f'(a)$ is defined, sequences that converge to $a$ that are eventually not constants are required. The continuum does the job. Homines sapientes have defined continuum, proved its properties, (The axiomatic construction of the continuum by itself provides its properties, thus "defined the continuum, proved its properties'' may not be considered a correct way of articulating the imagination behind the continuum. However, given the nature of this piece that it is a fable the author shall not be pedantic.) used it to understand the effect of small changes i.e., derivative, and have used it in real-life applications.
Imagining the continuum that cannot be seen, for example, no human
has possibly seen $x$ where $\pi=x$, and using the continuum for the advancement of homines sapientes possibly vindicates human ancestor's smirk.

A homo sapiens who could imagine something as incredible as the continuum would have murmured its properties in utter despair if the sapiens were unable to demonstrate the incredible imagination to other humans. The incredible news is that the sapiens did not suffer from such despair. In addition to the real numbers, the sapiens also has the access to another four incredible objects that are also created by the humans: paper, ink, pen, and letters which together produce the most incredible linguistics tool, in ways that are completely mysterious to the author, known as Proof'. By applying this tool every sapiens can demonstrate its imagination to any other fellow homo sapiens. Further, by definition of a proof, the fellow sapiens can verify that the proof provided by a sapiens by the means of which the sapiens claims to have demonstrated its imagination is indeed what the claim says it to be. Proof makes sure that a claim is not substantiated by any subjective interpretations. (A proof is defined within an axiomatic system, as before the author shall not be pedantic.) It maybe true that each lonely philosopher tiger can imagine axioms and deductions. However, if they cannot express their imaginations by writing proofs, then maybe it is natural to see tigers murmuring in despair. Thus, it maybe considered acceptable to say, "The invention of Proof, in particular, has made homo sapiens what it is now''.

It seems that home sapiens has transcended its ability to write proofs:
the homines sapientes have now come to possess the ability to create fairies. These fairies are called AI or artificial intelligence. These fairies can write and verify proofs written by humans. The concern is that these fairies, though they are manifestations of human imaginations, may start writing proofs more efficiently than humans. If the latter happens, then what is the point of Mathematicians? Number theorist Andrew Granville brings up this in a Quanta Magazine documentary, "When Computers Write Proofs, What's the Point of Mathematicians?''. Whether the fairies will surpass homines sapientes intelligence is a matter for another discussion. Whether, as articulated by Yuval Noah Harari, the presence of the fairies has ushered in the end of human history is a matter for a different discussion. (See Mustafa Suleyman & Yuval Noah Harari's debate: What does the AI revolution mean for our future?) But so far, it seems that homines sapientes have done incredibly well by inventing Proof, and thus may have atoned for the snooty ancestor's smirk.

We are changing!

Manjil Saikia — Mon, 04 Sep 2023 11:55:29 GMT

Gonit Sora (GS) started over 12 years ago (in 2011) when we (then very naive university students) got together to create a web-magazine publishing in English and Assamese catered to the student community in Assam. We did not dream big, but GS churned on its own and gradually it came to occupy a niche space in the Indian science and mathematics outreach space. But with the advent of several creators, both globally and in India, we feel that GS needs to change how it functions.

When we started out, our posts were about introducing simple mathematical concepts, news items and solved question papers of various Indian mathematical exams. All of these are dated now, and indeed we have removed much of that content from this website now. There are better venues now to find math news and math question papers of Indian exams. We will no longer cater to those needs.

We gradually ventured into doing interviews, some of which are quite well-done. All of the interviews have something which is unique with us, either the interviewee is not widely represented or they have told something which is not available in other interviews. For this and many other reasons, we have retained all of the interviews on the website. Sometimes we anticipated the importance of a person before others and this is also reflected in the interviews (for instance see the interview with TheLiverDoc who is now regularly in the news).

The old site contained many others things, including paid articles (whose content was always verified before we published those). Going forward we are no longer accepting any paid content on this site. We have mostly supported the website and its events from our own pockets, but there were some very generous individuals who donated to our cause in our initial days. We are thankful to them all, they know who they are!

We will post very few written articles from now on. What we will focus on is creating video and audio content. The videos will mostly be interviews and webinars as there are other much more popular and efficient video creators out there doing fantastic work which we cannot hope to emulate. However, we have a good professional network of mathematicians and scientists which we will leverage to bring to you specialized interviews and webinars. In fact, we ran a very successful webinar series during the initial days of the COVID-19 pandemic. We have so far not explored audio content but we plan to bring a monthly podcast series with scientists and mathematicians discussing their work at a professional level. So, please watch out for that.

In the past we did a few sporadic offline events, which we will streamline now and seek collaborations to implement them successfully. There are so many things that we planned to do, some of which never saw the light of day. This new avatar will try to finish some of those tasks.

One point of issue for which we apologize to our readers and patrons: during the migration of the old site into this new avatar, the mathematical symbols rendered using LaTeX were transformed into images and this has made some of the articles a bit difficult to read. We will try to fix this issue in the future, but for now some of the old articles are not very well-formatted.

If you liked what we did in the past, or have benefited from some of our content then please consider buying us a coffee so that we can keep the site running and create new content. You can get in touch with me directly to discuss any ideas, criticisms, or collaborations, please email me.

Interview with Dr. Cyriac Abby Philips on Evidence Based Medicine

Manjil Saikia — Mon, 23 May 2022 03:03:29 GMT

Gonit Sora recently did a live interview with Dr. Cyriac Abby Philips where we discussed evidence based medicine, his work on liver toxicity due to herbal products, scientific literacy and much more.

Dr. Philips is currently working in the Clinical and Translational Hepatology & Monarch Liver Laboratory, The Liver Institute, Center of Excellence in Gastrointestinal Sciences, Rajagiri Hospital. He is also an Expert Member, Center of Excellence in Microbiome Studies, Kerala Development and Innovation Strategic Council (K-DISC), Ministry of Health and Family Welfare, State Government of Kerala

Dr. Philips is a two-time American Association for the Study of Liver Diseases (AASLD) clinical hepatology plenary awardee (2015, 2017) and three-time AASLD Young Investigator Award winner (2015, 2016, 2017) and the 2017 winner of the Young Investigator Award of the European Association for the Study of Liver (EASL). He was conferred the President of India Gold in Hepatology at the Institute of Liver and Biliary Sciences, New Delhi in 2016.

He has published several peer reviewed articles in high impact journals such as the New England Journal of Medicine, Journal of Hepatology, Hepatology and American Journal of Gastroenterology. He is currently an editorial board member of the Indian Journal of Gastroenterology, Journal of Clinical and Translational Hepatology, BMC Gastroenterology and Biomed Research International.

The full interview is uploaded on our YouTube channel, as well as below.

Introduction to Machine Learning

Priyakshi Goswami — Mon, 22 Nov 2021 10:30:00 GMT

Machine Learning is the buzzword of the moment in computer science and technology and undeniably one of the most powerful technologies today. Machine learning is an integral part of many applications we use every day such as Google search, Google maps, Netflix recommendations, spam email detection, virtual assistants like Alexa and Siri, and even medical diagnosis and healthcare. There are very few domains that are untouched by this rapidly expanding field.

What is Machine Learning?

A machine is said to learn from experience E with respect to some task T and some performance measure P, if its performance on T, as measured by P, improves with experience E.
– Tom Mitchell
Machine Learning, 1997.

Machine Learning is a subset of Artificial Intelligence that develops systems which can automatically learn and improve from experience without being explicitly programmed. In simple words it is the process of training a piece of software, called a model, to make useful predictions using a data set.

The basic premise of machine learning is to build models from example inputs to make data-driven predictions or decisions. The world is filled with an enormous amount of data today which continues to grow. This massive amount of data is useless unless we learn how to analyze them. Machine Learning helps us give meaning to this data. It converts the information in the data into knowledge.

Major approaches in Machine Learning

The famous No Free Lunch theorem in machine learning states that there is no single algorithm that will work well for all tasks. Each task that you try to solve has its quirks. Hence, there are a lot of algorithms and approaches to suit each one of them.

Machine Learning is typically classified into three major paradigms – supervised, unsupervised and reinforcement learning. Although not often defined as a fourth class of machine learning, there is another approach called semi-supervised learning which represents a middle ground between supervised and unsupervised learning. These different types of machine learning algorithms differ in their approach, the type of data they input and output, and the type of problems that they are intended to solve.

Supervised Learning

Supervised Learning is the most commonly used branch of machine learning. Typically every new machine learning practitioner starts with supervised learning algorithms. With supervised learning you use labelled data, which is a data set that has been classified, to infer a learning algorithm. This data is called training dataset. This training data is used as the basis for predicting the classification of other unlabelled data (also known as test data set) through the use of machine learning algorithms. So, the basic goal is to find the mapping function to map the input variable (X) with the output variable (Y) in the training data.

Y = f(X)

One practical example of supervised learning problems is predicting house prices. For this, we first need data about he houses: square foot area, number of rooms, whether a house has a garden or not, and other such features. We then need to know the housing price, i.e. the target labels. By leveraging data coming from thousands of houses, we can now train a supervised machine learning model to predict a new house’s price based on the examples observed by the model.

Datasets are generally split into training, validation, and testing datasets before using them. Models will always perform optimally on the data they are trained on. Being able to adapt to new inputs and make predictions is the crucial generalisation part of machine learning. Over-training the model often results in the model’s inability to adapt to new, previously unseen data, which is referred to as overfitting.

Methods of supervised learning – Classification and Regression

The output or the target feature in a supervised machine learning model could be a number, in which case it is known as a regression model; or alternatively could be a category and in this case it is called a classification model. The housing price example we discussed above is a regression problem. A simple widely used classification algorithm is email spam classification based on spam/non-spam email examples.

Unsupervised Learning

Today’s machine learning applications need a lot of labelled data to have good performance, but most of the world’s data is not labelled. For machine learning to advance, algorithms will need to learn from unlabelled data and make sense of the world from pure observation, much like how children learn to operate in the real world after birth without too much guidance.

Most of human and animal learning is unsupervised learning. If intelligence was a cake, unsupervised learning would be the cake, supervised learning would be the icing on the cake, and reinforcement learning would be the cherry on the cake.
-Yann LeCun

Unsupervised learning is about extracting underlying structure and detecting patterns from an unlabelled dataset without any reference to labelled outcomes or predictions. These algorithms are named unsupervised because the patterns that may or may not exist in a dataset are not informed by a target and are left to be determined by the algorithm. Unlike supervised learning, here only the input data is provided; we lack the access to labelled examples.

When presented with data, an unsupervised machine will search for similarities between the data, namely images, and separate them into individual groups, attaching its own labels onto each group. This kind of algorithmic behaviour is very useful when it comes to segmenting customers as it can easily separate data into groups without any form of bias that might hinder a human due to pre-existing knowledge about the nature of the data on the customers.

Methods of Unsupervised Learning

Some of the main methods used in unsupervised learning are cluster analysis, association rules and dimensionality reduction.

Cluster analysis, also known as data segmentation, involves grouping or segmenting a collection of objects into subsets or “clusters”. This involves successively grouping the clusters themselves so that at each level of the hierarchy, clusters within the same group are more similar to each other than those in different groups.

Association rule mining finds interesting associations and relationships among large sets of data items. This rule shows how frequently an itemset occurs in a transaction. A typical example is Market Based Analysis.

Large numbers of input features often make a predictive modelling task more challenging to model. This is often referred to as the curse of dimensionality. Dimensionality reduction techniques reduce the number of input variables in a dataset to simplify a classification or a regression problem.

Semi-supervised Learning

The most inherent disadvantage of any supervised learning algorithm is that the dataset needs to be hand-labelled either by a Machine Learning Engineer or a Data Scientist. This is a very costly process, especially when dealing with large volumes of data. On the other hand, the main disadvantage of unsupervised learning is that it’s application spectrum is limited. To counter these disadvantages, the concept of semi-supervised learning was introduced.

In semi-supervised learning, the algorithm trains on a combination of a small amount of labelled data and a large amount of unlabelled data. The basic idea is to first cluster the similar data using an unsupervised learning algorithm and then use the existing labelled data to label the rest of the unlabelled data.

Semi-supervised learning models are becoming widely applicable in scenarios across a large variety of industries. Text document classifiers are a classic example of semi-supervised learning. Semi-supervised learning is ideal in this situation because it would be nearly impossible to find a large amount of labelled text documents. This is simply because it is not time efficient to have a person read through entire text documents just to assign it a simple classification. So, the algorithm learns from a small amount of labelled text documents while still classifying a large amount of unlabelled text documents in the training data.

Reinforcement Learning

Reinforcement Learning is one of the hottest research topics currently. This machine learning method is quite different from both supervised and unsupervised learning. It doesn’t use labels and instead uses rewards to learn. Reinforcement learning is a type of machine learning technique that enables an agent to learn by directly interacting with its environment through trial and error using feedback from its own actions and experiences.

To get the machine to do what the programmer wants, the artificial intelligence gets either rewards or penalties for the actions it performs. The learner is not told which actions to take, but instead it must discover which actions yield the maximum reward by trying them.

Markov Decision Processes (MDPs) are mathematical frameworks typically used to describe an environment in reinforcement learning. An MDP consists of a set of finite environment states S, a set of possible actions A(s) in each state, a real valued reward function R(s) and a transition model P(s’, s | a). However, real world environments are more likely to lack any prior knowledge of environment dynamics. In such cases model-free RL methods like Q-learning and SARSA (State-Action-Reward-State-Action) are used.

Fig 1 – Agent-Environment interaction in machine learning [Source]

One of the most incredible things about reinforcement learning is that it brings machine learning closer to how we humans learn. Throughout our lives, interactions are undoubtedly a major source of knowledge about the environment and ourselves. Whether it be learning to drive a car or to hold a conversation, we are aware of how our environment responds to what we do, and seek to influence what happens through our behaviour.

Games are very popular in RL research. DeepMind’s AlphaGo Zero was able to learn the game of Go from scratch using reinforcement learning and end up achieving superhuman performance. It learned by playing against itself. After 40 days of self-training, Alpha Go Zero was able to outperform the version of AlphaGo known as Master that defeated world number one Ke Jie in what is often referred to as one of the most complex board games in human history.

Training models that control autonomous cars is another excellent example of a potential application of reinforcement learning. Apart from this, RL also has applications in industry automation, trading and finance, natural language processing(NLP), healthcare, recommendation systems

Mathematics and Machine Learning

For all the operations we perform on data, there is one common foundation that helps us achieve all of this through computation – and that is Mathematics. Despite the immense possibilities of Machine and Deep Learning, a thorough mathematical understanding of many of these techniques is necessary for a good grasp of the inner workings of the algorithms and getting good results. Mathematics defines the underlying concept behind the algorithms and tells which one is better and why.

The major topics of Mathematics widely used in ML are Linear Algebra, Statistics, Probability Theory and Calculus.

Linear Algebra is a key foundation to the field of machine learning. It focuses mainly on computation. Some of the Machine Learning algorithms like Linear regression, Logistic regression, SVM and Decision trees use Linear Algebra in building the algorithms. Calculus deals with optimizing the performance of machine learning algorithms. Statistics plays an important role in the field of Machine Learning as it deals with large amounts of data. It deals with the statistical methods of collecting, presenting, analyzing and interpreting the numerical data.

Probability forms the basis of sampling. In machine learning, uncertainty can arise in many ways – for example – noise in data. Probability provides a set of tools to model uncertainty. Probability also forms the basis of specific algorithms like Naive Bayes classifier. Many iterative machine learning techniques like Maximum likelihood estimation (MLE) are based on probability theory. MLE is used for training in models like linear regression, logistic regression and artificial neural networks. Apart from these, model evaluation techniques require us to summarize the performance of a model based on predicted probabilities.

Conclusion

This article is a brief introduction to a few basic widely used machine learning algorithms. Another very important subfield of machine learning that is not discussed in the article is Deep Learning. Deep Learning consists of machine learning techniques based on Artificial Neural Networks. The introduction and advancement of deep learning has resulted in a revolution in the applications of machine learning in recent years.

2021 Nobel Prize in Physics

Abhigyan Ray — Sat, 09 Oct 2021 17:19:53 GMT

The Nobel Prize in Physics 2021 was awarded “for groundbreaking contributions to our understanding of complex systems” with one half jointly to Syukuro Manabe and Klaus Hasselmann “for the physical modeling of the Earth’s climate, quantifying variability and reliably predicting global warming” and the other half to Giorgio Parisi “for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales.”

Syukuro Manabe received his Ph.D. at the University of Tokyo in 1958. He then headed to the U.S. to work at the U.S. Weather Bureau until 1997. He moved back to Japan to work at the Frontier Research System for Global Change, serving as director of the Global Warming Research Division, and in 2002 he returned to the U.S. to Princeton University, where he is currently a senior meteorologist. Klaus Hasselmann completed his Ph.D. in physics from the University of Göttingen in 1957. Next, he moved to the Institute of Naval Architecture at the University of Hamburg, where he remained until 1961. He then went to the U.S. to work at the Scripps Institution of Oceanography before moving back in 1964 to the University of Hamburg. In 1975 he became a director of the Max-Planck Institute of Meteorology in Hamburg, eventually retiring in 1999. Giorgio Parisi graduated in 1970 with a Ph.D. in high-energy physics from La Sapienza University. He worked at the Laboratori Nazionali di Frascati and the University of Rome Tor Vergata before returning in 1992 to La Sapienza, where he serves as a Professor.

This year’s Nobel Prize in Physics focuses upon the complexity of physical systems, from the largest scales experienced by humans, such as earth’s climate, down to the microscopic structure and dynamics of mysterious and yet commonplace materials, such as glass.

During the 1950s, with the advent of modern-day computing, large-scale numerical weather forecasting originated at the Institute for Advanced Study in Princeton, led by the legendary John von Neumann himself. The study of theoretical and experimental aspects of atmospheric and oceanic dynamics (geophysical fluid dynamics–GFD) was carried out in parallel. Numerous pioneers were recruited, including Syukuro Manabe in 1959 by Joseph Smagorinsky, head of the U.S. Weather Bureau’s General Circulation Research Laboratory, which later moved to Princeton to become the Geophysical Fluid Dynamics Laboratory. Over half a century earlier, another Nobel laureate, Svante Arrhenius, had conceptualized the concept of warming of the earth’s atmosphere through the greenhouse effect. It built the scientific edifice central to the atmospheric column models used in successively more complex treatments that have developed since then.

In 1967 Syukuro Manabe and Richard Wetherald published the first computer model of climate sensitivity to fluctuating atmospheric carbon dioxide levels. To approximate the climate, they simulated a single column of air and looked at how convection told the story of varying temperatures. Manabe resoundingly demonstrated how increased levels of carbon dioxide in the atmosphere lead to increased temperatures at the earth’s surface and was the first person to explore the interaction between radiation balance and the vertical transport of air masses. Manabe helped showcase that the observations show relatively slight seasonal variation in the climatological latitudinal relative humidity profiles in the northern hemisphere. In contrast, the absolute humidity (saturation vapor pressure) will depend sensitively on temperature, leading to the critical result of Manabe and Wetherald, their calculation of climate sensitivity of 2.3 degrees Celsius warming.

A decade later, in 1976, Klaus Hasselmann demonstrated that the climate responded to random variability by creating a model that linked together weather and climate, thus answering the question of why climate models can be reliable despite weather being changeable and chaotic. Hasselmann creatively utilized fundamental physics concepts to quantify the surface ocean wave spectra. Building on basic concepts in turbulence and Lorenz’s work on chaos, he derived a generalizable stochastic description of ocean climate in which the “noise” is associated with the “weather.” Hasselman also developed methods for identifying specific signals, fingerprints that natural phenomena and human activities imprint in the climate, helping climate scientists characterize how much warming was truly anthropogenic. Hasselmann provided a statistical roadmap for hundreds of subsequent climate change detection and attribution studies that eventually provided solid scientific support for the stark conclusion reached by the IPCC in 2013: “It is extremely likely that human influence has been the dominant cause of the observed warming since the mid-20th century.“

Giorgio Parisi is a true giant of theoretical physics who has been honored for his groundbreaking work in the area of complex and disordered systems and their related fluctuations. During the early 1970s, Parisi began to work on the theory of phase transitions within solids, mainly focussing on the usage of field theory techniques from high-energy physics to condensed-matter physics. He conceptualized some of the most potent theoretical machinery for understanding the behavior of spin glasses and other complex systems. Spin glasses are disordered magnetic systems that appear to have a phase transition to a state where each magnetic atom is stably aligned. Still, the alignment direction varies randomly between atoms, resulting in “frustrated” interactions due to a disorder, which gives them a rich set of behaviors. Parisi realized that in contrast to ferromagnets which have only two “pure states” (up/down) in the ordered phase, there must be an infinite number within the ordered phase of the spin glass, thereby not only providing the solution but opening up a rich array of extensions to a wide range of spin-glass and other systems. Wielding advanced mathematics, he thus solved the problem of replica symmetry breaking. Later, Parisi with Mezard and Virasoro significantly clarified the physical meaning of the mysterious mathematics involved in this scheme in terms of the probability distribution of overlaps and the ultrametric structure of the configuration space.

Parisi is a legendary theorist whose work has had a tremendous impact on various other fields, including particle physics, theoretical computer science, quantum field theory, neuroscience, evolutionary biology, immunology, and statistical mechanics. Alongside the replica method for analyzing the Sherrington-Kirkpatrick model, Parisi has made uniquely wide-ranging contributions to many areas of physics and other sciences that have impacted and inspired generations of scientists. They include the study of scaling violations in deep inelastic processes (Altarelli-Parisi equations), the proposal of the superconductor’s flux confinement model as a mechanism for quark confinement enabling very large-scale simulations of QCD, the use of supersymmetry in classical statistical systems, the introduction of multifractals in turbulence, the study of idiotypic network theory for antibodies in theoretical immunology, the stochastic differential equation for growth models for random aggregation (the Kardar-Parisi-Zhang model), scale-free correlations in starling flocks, random constraint satisfaction problems, identifying universal behavior in the dynamics of how people register for conferences.

With Parisi, the Physics Nobel Committee seems to be continuing its recent tradition of honoring a great theorist alongside two experimentalists as the examples of Peebles and Penrose in the preceding year’s showcase. It is a much welcome change for an institution that never really gave theorists their fair share until late. As arguably the most well-known scientific prize that enraptures the attention of a worldwide audience, allowing legendary theorists on the bully pulpit helps drive attention to their pivotal role in germinating ideas that go on to revolutionize their fields and world at large. This is vital at a time when myopic governments and incompetent administrators, in their quest to focus on next generation technologies, quixotically end up shutting down pure mathematics and theoretical physics departments, unable to grasp their fundamental importance in seeding the basic science behind those revolutionary technologies. As Manabe proclaimed about the grim impacts of climate change that are coming to fruition now, “I did these experiments out of pure scientific curiosity. I never realized that it would become a problem of such wide-ranging concern for all of human society.”

One of the biggest pantomimes of science is that it is apolitical. This ridiculous notion has been debunked time and time again, yet it continues to persist in some form or the other. The pandemic has expressly showcased how science can be misused, with terrible consequences, by politicians to further their divisive agenda during a public health crisis. Moreover, the past few decades have further displayed how politicians and groups with vested interests have been in cahoots to play down the devastating effects of climate change and global warming by resorting to spurious arguments like the science not being perfect and decisive. As the world faces a reckoning moment, with most of the dire predictions transpiring sooner than later, the Nobel committee decided to take the bull by its horns and brushed aside these fallacious arguments when announcing this year’s prize.

Thors Hans Hansson, chair of the Nobel Committee for Physics, vociferously thundered during the announcement, “The discoveries being recognized this year demonstrate that our knowledge about the climate rests on a solid scientific foundation, based on rigorous analysis of observations. This year’s Laureates have all contributed to us gaining deeper insight into the properties and evolution of complex physical systems.” Asked during the press conference if the Nobel committee was sending a message to world leaders with the award, Göran Hansson, secretary-general of the prize-awarding Royal Swedish Academy of Sciences, emphatically replied: “What we are saying is that the modeling climate is solidly based in physical theory and solid physics. Global warming is resting on solid science. That is the message.” And the great Parisi, no stranger to calling out lackluster funding of basic research, earnestly insisted that in the upcoming 26th United Nations Climate Change Conference at Glasgow in November, “It’s very urgent that we take a very strong decision and move at a very strong pace. We are in a situation where we have positive feedback and an accelerating increase of temperature. For the future generations, we have to act now in a very fast way.”

Finally, for a world that has developed a very vague understanding of scientific models during the pandemic, the Nobel committee presciently observed, “Recognizing the work of this troika reflects the importance of understanding that no single prediction of anything can be taken as inviolable truth, and that without soberly probing the origins of variability we cannot understand the behavior of any system. Therefore, only after having considered these origins do we understand that global warming is real and attributable to our own activities, that a vast array of the phenomena we observe in nature emerge from an underlying disorder, and that embracing the noise and uncertainty is an essential step on the road towards predictability.” In these wise words lies the true essence of science.

Nobel Prize in Medicine 2021

Abhigyan Ray — Wed, 06 Oct 2021 00:56:00 GMT

The Nobel Assembly at the Karolinska Institute has decided to award the 2021 Nobel Prize in Physiology or Medicine to David Julius and Ardem Patapoutian “for their discoveries of receptors for temperature and touch”. David Julius is professor and chair of the department of physiology at the School of Medicine in the University of California, San Francisco and a trustee of the Howard Hughes Medical Institute (HHMI). Ardem Patapoutian is a HHMI investigator in the department of neuroscience at Scripps Research in California as well as an adjunct professor in the neuroscience program of the University of California, San Diego.

One of the most innate human qualities is the sense of touch. A warm embrace with a loved one or an enthusiastic high five after winning a game is an experience humanity shares across borders and cultures. Yet for something so intrinsic to being human, the molecular and physicochemical underpinnings of the sensation of touch eluded us before today’s Nobel laureates set out to crack open the problem “for the greatest benefit of humankind” and win the very first prize in the field of sensory transduction. Their transformative discovery of receptors for temperature and pressure revolutionized the field of neuroscience by providing a molecular and neural basis for thermosensation and mechanosensation.

After training under a stellar lineage of Nobel winning academics, Julius started his own lab at UCSF in the early 90’s trying to gain an understanding of how signals responsible for temperature sensation and other sensory phenomena are transmitted by the nervous system. In this quest, he exploited distinctive molecules from the natural world, like the capsaicin molecule and tarantula toxins, to elucidate the molecular mechanisms of pain sensation. Julius’ group created a library of millions of DNA fragments corresponding to genes that are expressed in the sensory neurons which can react to pain, heat and touch. They then plugged genes from this collection into cells that do not normally react to capsaicin to find the single gene that caused the sensitivity. Using capsaicin (8-methyl-N-vanillyl-6-nonenamide), the compound in chili peppers that elicits the sensation of heat, they identified the gene encoding the first temperature sensor, the ion channel TRPV1 “heat” receptor. They went on to discover that TRPV1 is activated by high temperature, high concentrations of protons found in ischemic tissues and chemical compounds generated during inflammation, thus providing a molecular integrator for both thermosensation and inflammatory signals. Similar results were obtained with the detection of TRPM8 “cold” receptor in menthol, the cooling agent in mint leaves that evokes an icy cool sensation, and also with regards to the pungency of wasabi. The seminal discovery of TRPV1 initiated an intense investigation that has strongly established the critical role of a family of TRP channels for thermal sensation and conclusively showcase that several TRP channels gated at different temperature ranges act together to code for temperature and heat-induced pain in the somatosensory nervous system.

As Julius’ seminal work was making waves in top journals like Nature, an Armenian immigrant fleeing the war in Lebanon who had come to the USA to become a doctor but quickly “fell in love with research”, also began investigating the molecular bases of sensory perception. After briefly coinciding with Julius at UCSF during his postdoc, Ardem Patapoutian went on to found his own group at Scripps Research Institute that seeked to gain an understanding of the intricacies of physicochemical transmission of physical stimuli such as temperature and pressure. Patapoutian’s lab used a functional screen of candidate genes expressed in a mechanosensitive cell line to identify a family of ion channels activated by mechanical stimuli and formed the basis of how we sense touch, pain, sound, and blood flow. Two mechanically-activated cation channels, named PIEZO1 and PIEZO2 (derived from the Greek word for pressure – piezos)), were identified and shown to represent an entirely novel class of ion channels functioning as mechanical sensors. Genetic studies established that Piezo2 is the principal mechanical transducer for touch, proprioception, and lung stretch, and that Piezo1 mediates blood-flow sensing, which impacts blood pressure regulation and vascular development. Additionally, the Patapoutian lab co-identified SWELL1 (LRRC8A), an ion channel critical for regulating cell volume in response to osmotic shock. They further demonstrated that Piezos forms pressure-sensing channels and that they are directly responsible not only for pressure sensing in the skin by Merkel cells, proprioceptors and touch sensory terminals but also to sense pressure by nerve terminals in blood vessels and in the lungs.

Thanks to the untiring efforts of these pioneering researchers and their seminal discoveries, that has been the subject of this year’s Nobel Prize, we have been able to develop a finer understanding of mechanobiology in health and disease. Today we have a molecular and neural basis for thermosensation and mechanosensation which further enables us to apply it to multiple domains, right from devising ways to reduce chronic and acute pain associated with a range of diseases to better mechanisms to control blood pressure. We have gained insights into the mechanisms of proprioception that can help us treat neurodegenerative diseases that result in loss of balance and motor control, and are continuing to hunt for novel mechanosensors that affect red blood cell volume, vascular physiology, and underlie a broad range of human genetic disorders. Wielding an array of genetic, electrophysiological, and behavioral methods, researchers are now unravelling the contribution of the ion channels to detecting heat and cold as well as their activity modulation in response to tumor growth, infection, or other forms of injury that produce inflammation and pain hypersensitivity alongside the role of specific neurotransmitter receptors, such as those activated by serotonin or extracellular nucleotides, in physiological and behavioral processes, such as feeding, anxiety, pain, thrombosis, and cell growth and motility.

This year’s Nobel Prize to a phenomenal set of neuroscientists, celebrates the great importance of curiosity driven fundamental research in the basic sciences. At a time when it is easy to get swayed by the hype over supposedly revolutionary technologies, it is important to remember that decades of basic research in the labs of folks like Julius and Patapoutian is what drives these innovations in the first place. Similarly, the vaccines for COVID19 might have been developed and got out into the market in well under a year’s time but the underlying technology like mRNAs and lipid nanoparticles have been refined and studied for well over a couple of decades. The groundbreaking work in the labs of these two pioneers, Julius and Patapoutian, was equally strenuous and spanned three decades of laborious efforts and relentless self belief by the researchers in their labs. Identification of the cellular target of capsaicin in the former’s lab required painstakingly sifting through a cDNA library from sensory neurons in a functional screen to look for a gene that could confer capsaicin sensitivity to cells that were normally unresponsive. In the latter’s lab too, the discovery of the gene encoding the receptors that enables nerve cells to sense pressure required a cell line in a petri dish that reacted to being poked by a micropipette with an electrical signal for which they went through 72 candidate genes before identifying the one that rendered the cells impervious to the pipette’s touch.

Furthermore, the laureates overcame several economic and societal impediments to attain the pinnacle.David Julius’s grandparents fled European anti-Semitism and moved to New York City and Ardem Patapoutian grew up in Lebanon, to an already displaced family that had escaped the Armenian genocide, and then immigrated to the US for educational opportunities as the situation spiralled out of control back at home. As the world turned more inward looking over the past decade due to the shameless demagoguery of elected authoritarians, it is prudent to remember the personal backgrounds of these laureates that serve to showcase that society progresses best when it offers a warm and welcoming place to everyone regardless of trivial distinguishing factors like colour, gender, creed, and ethnicity. Additionally, at a time when funding cuts are rampant for basic science research, with entire pure science departments shutting down on the whims of inept administrators, its important that as we celebrate the outstanding research of the newly minted Nobel Laureates on an innate human sensation, we should also raise a toast to the fundamental role curiosity driven basic research has played in advancing our society since time immemorial.

Emmy Noether faced sexism and Nazism – 100 years later her contributions to ring theory still influence modern math

Manjil Saikia — Mon, 16 Aug 2021 18:10:25 GMT

Emmy Noether made significant contributions to theoretical mathematics. Konrad Jacobs, Erlangen/Wikimedia Commons, CC BY-SA

Tamar Lichter Blanks, Rutgers University

When Albert Einstein wrote an obituary for Emmy Noether in 1935, he described her as a “creative mathematical genius” who – despite “unselfish, significant work over a period of many years” – did not get the recognition she deserved.

Noether made groundbreaking contributions to mathematics at a time when women were barred from academia and when Jewish people like herself faced persecution in Nazi Germany, where she lived.

The year 2021 marks the 100th anniversary of Noether’s landmark paper on ring theory, a branch of theoretical mathematics that is still fascinating and challenging mathematicians like me today.

I remember the first time I learned about Noether and the surprise I felt when my professor referred to the brilliant ring theorist as “she.” Even though I am a woman doing mathematics, I had assumed Noether would be a man. I was surprised at how moved I was to learn she was a woman, too.

Her inspiring story is one that not many people know.

A rare woman in mathematics

Emmy Noether earned a doctorate in mathematics in 1909, but women were not allowed to work as professors at that time in Germany. Mathematical Assoc, iabutit onat ofth Aatme triimcae, v ia WikimediaCommons

Noether was born in 1882 in Erlangen, Germany. Her father was a math professor, but it must have seemed unlikely to a young Noether that she would follow in his footsteps. At the time, few women took classes at German universities, and when they did they could only audit them. Teaching at a university was out of the question.

But in 1903 – a few years after Noether graduated from a high school for girls – Erlangen University started to let women enroll. Noether signed up and eventually earned her doctorate in mathematics there.

That doctorate should have been the end of her mathematical career. At the time, women were still not allowed to teach at universities in Germany. But Noether stuck with mathematics anyway, staying in Erlangen and unofficially supervising doctoral students without pay. In 1915, she applied for a position at the prestigious University of Göttingen. The dean at the university, also a mathematician, was in favor of hiring Noether, although his argument was far from feminist.

“I think the female brain is unsuitable for mathematical production,” he wrote, but Noether stood out as “one of the rare exceptions.”

Unfortunately for Noether, the Prussian Ministry of Education would not give the university permission to have a woman on their faculty, no matter how talented. Noether stayed in Göttingen anyway and taught courses listed under the name of a male faculty member.

During those years, she kept doing research. While she was still an unofficial lecturer, Noether made important contributions to theoretical physics and Einstein’s theory of relativity. The university finally granted her lecturer status in 1919 – four years after she applied.

The University of Göttingen, seen here, was unable to hire Noether as a professor, so she taught courses under a male colleague’s name. Daniel Schwen/WikimediaCommons, CC BY-SA

A revolution in ring theory

In 1921, only two years after becoming an official lecturer, Noether published revolutionary discoveries in ring theory that mathematicians are still pondering and building upon today. Noether’s work in ring theory is the main reason that I, like many mathematicians today, know her name.

Ring theory is the study of mathematical objects called rings. Despite the name, these rings have nothing to do with circles or ring-shaped objects – theoretical or otherwise. In mathematics, a ring is a set of items you can add, subtract and multiply and always get another object that is in the set.

A classic example is the ring known as Z. It is made of all the integers – positive and negative whole numbers like 0, 1, 2, 3, -1, -2, -3 and so on – and it is a ring because if you add, subtract or multiply two integers, you always get another integer.

There are infinitely many rings, and each one is different. A ring can be made of numbers, functions, matrices, polynomials or other abstract objects – as long as there’s a way to add, subtract and multiply them.

One reason rings are so interesting to mathematicians is that often it is possible to tell something is a ring, but it’s difficult to know much about the specifics of that particular ring. It’s like seeing a croissant at a fancy bakery. You know you are looking at a croissant, but you might not know whether it’s filled with almond paste, chocolate or something else altogether.

Instead of focusing on one ring at a time, Noether showed that a whole class of easy-to-identify rings all share a common internal structure, like a row of houses with the same floor plan. These rings are now called Noetherian rings, and the structure they share is like a map that guides the mathematicians who study them.

Noetherian rings show up all the time in modern mathematics. Mathematicians still use Noether’s map today, not just in ring theory, but in other areas such as number theory and algebraic geometry.

A plaque in her home town of Erlangen honors Emmy Noether and mentions her immigration to the U.S. Norman Rönz/WikimediaCommons, CC BY-SA

Escape from Nazi Germany

Noether published her famous ring theory paper and other important results in mathematics while she was a lecturer in Göttingen from 1919 to 1933. But in the spring of 1933, the University of Göttingen received a telegram: Six faculty members – including Noether – had to stop teaching immediately. The Nazis had passed a law barring Jews from professorship.

Noether’s response, it seems, was calm. “This thing is much less terrible for me than it is for many others,” she wrote in a letter to a fellow mathematician. But she was out of a job, and no university in Germany could hire her.

Help came from the United States. Bryn Mawr, a women’s college in Pennsylvania, offered Noether a professorship through a special fund for refugee German scholars. Noether accepted the offer and, as a professor at Bryn Mawr, she mentored four younger women – one doctoral student and three postdoctoral researchers – in advanced mathematics.

Noether’s time at Bryn Mawr was, tragically, short. In 1935 she had surgery to remove a tumor and died unexpectedly four days later.

At Noether’s funeral, mathematician Hermann Weyl compared her sudden passing to “the echo of a thunderclap.” In her short life, Noether shook up mathematics. She kept teaching and learning even when women and Jews were not welcome. One hundred years later, her mathematical genius and “unbreakable optimism” are qualities to admire.

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Tamar Lichter Blanks, PhD Candidate in Mathematics, Rutgers University

This article is republished from The Conversation under a Creative Commons license. Read the original article.

Mathematics of the Football

Manjil Saikia — Mon, 14 Jun 2021 15:03:22 GMT

Prof. Fernando Rodriguez Villegas (ICTP, Trieste) gave a talk on 8th May, 2021 on “Mathematics of the Football”.

The full video of the talk is available in our YouTube channel as well as below.

A list of all past and future webinars are available at this link.

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What is the Riemann Hypothesis, and why does it matter?

Manjil Saikia — Wed, 21 Apr 2021 19:38:29 GMT

Prof. Ken Ono (University of Virginia, USA) gave a talk on 21st April, 2021 on “What is the Riemann Hypothesis, and why does it matter?”.

The full video of the talk is available in our YouTube channel as well as below.

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ACM Turing Award 2020

Abhigyan Ray — Thu, 15 Apr 2021 15:08:35 GMT

Alfred Vaino Aho and Jeffrey David Ullman were declared the recipients of the 2020 ACM A.M. Turing Award “for fundamental algorithms and theory underlying programming language implementation and for synthesizing these results and those of others in their highly influential books, which educated generations of computer scientists.” Alfred V. Aho is the Lawrence Gussman Professor Emeritus of Computer Science at Columbia University and Jeffrey D. Ullman is the Stanford W. Ascherman Professor Emeritus of Computer Science at Stanford University. The Association for Computing Machinery (ACM) A.M. Turing Award, christened the “Nobel Prize of Computing,” is named in honour of Alan Turing, the British mathematician, who articulated the mathematical foundation and the limits of computing. The Turing Award recognizes contributions that have a lasting impact on computing and it is ACM’s most prestigious technical award.

Alfred Vaino Aho

Aho and Ullman earned their doctoral degrees at Princeton University, a year apart before joining the famed Bell Labs, where they worked together from 1967 to 1969. During this short period, they kickstarted a collaboration that would last a lifetime and revolutionise the world at large. Their early innovative research in formal languages and compiler theory led to key algorithms for modern compilers and string-pattern matching tools. Compilers are programs that translate source code from a high-level programming language to a lower level language (assembly language, object code, or machine code), and thereby help create an executable program.

Aho and Ullman made a wide range of contributions to programming language compilers, formal language theory and invented efficient algorithms for lexical analysis, syntax analysis, code generation, code optimisation, text processing applications, and programming language translation. They helped build the YACC parser-generator, a staple method for quickly building parsers for programming languages, by making Knuth’s LR(k) parsing algorithm work with simple grammars that technically did not meet the requirements of an LR(k) grammar. Their code generation algorithms influenced the design of retargetable C compilers, which further facilitated the porting of the UNIX operating system from minicomputers to supercomputers.

Jeffrey David Ullman

Ullman was one of the founders of database theory and made crucial contributions to data mining and data integration. He was instrumental in developing a theory of algorithms for the modern “MapReduce” style of parallel programming. He pioneered the field of join processing in map-reduce environments that has had tremendous impact on big data processing in parallel and distributed architectures. Aho built popular and fundamental Unix tools like Lex and grep. He also invented the Aho–Corasick algorithm, alongside Margaret Corasick. It is a string-matching algorithm that constructs an automation for finding words in an input string that are contained in a dictionary. Aho and Corasick collaborated with Peter Weinberger and Brian Kernighan, to create the AWK programming language, a scripting language that begated dynamic, “problem-focused” languages like Perl, Python, Javascript, and Ruby.

Ullman and Aho have also been brilliant expositors and have authored terrific textbooks that have been the Magna Carta in their fields. They penned the definitive treatise on compiler technology, Principles of Compiler Design (1977) and nicknamed the “Dragon Book” for its colourful cover. It beautifully elucidates the phases in translating a high-level programming language to machine code, modularizing the entire enterprise of compiler construction. It still remains the prescribed textbook on compiler design. Ullman also penned the Principles of Database Systems (1988), a comprehensive description of the theory and principles of database systems. Along with another Turing Awardee, John Hopcroft, Ullman and Aho wrote the classic, The Design and Analysis of Computer Algorithms (1974), which is a standard textbook for algorithms courses throughout the world, when computer science was still in its infancy.

Ullman left Bell Labs after three years and was part of the faculty at Princeton and Stanford while Aho plied his trade at Bell Labs for over three decades before transitioning to conventional university academia. The pioneering work they carried out at Bell Labs is reminiscent of a glorious time when industry heartily funded truly blue-sky research. Till date, Bell Labs has 9 Nobel Prizes and 5 Turing Awards, to its credit.

Both Ullman and Aho have built an illustrious body of research in a wide array of computer science disciplines like automata, formal languages, language theory, compilers, data structures, algorithms, database theory, and graph theory They have also published numerous research papers that have garnered thousands of citations and authored books that have trained generations of computer scientists. They have been admitted to the ranks of many esteemed societies, academies, fellowships, and have earned numerous accolades like the IEEE John von Neumann Medal “for outstanding achievements in computer-related science and technology.” Their pioneering contributions, both individually and jointly, have helped shape and power the modern day world. Their theory underlies the programs that enable instantaneous internet search results and Internet of Things (IoT) ecosystem.

The prize is not without its controversies and Ullman has been accused of holding xenophobic and discriminatory views against students and researchers from Iran. The links here, here and here contain a better description of the nuances of the allegations against Ullman. This also reiterates the fact that prizes, no matter how prestigious, don’t serve as tests of character.

Laszlo Lovasz and Avi Wigderson share the 2021 Abel Prize

Abhigyan Ray — Sat, 20 Mar 2021 23:24:05 GMT

Last time the phrase “theoretical computer science” found mention in an Abel Prize citation was in 2012 when the legendary Hungarian mathematician, Endre Szemerédi was awarded mathematics’ highest honour. During the ceremony, László Lovász and Avi Wigderson were there to offer a primer into the majestic contributions of the laureate. Little did they know, nearly a decade later, they both would be joint recipients of the award. The Norwegian Academy of Science and Letters has awarded the 2021 Abel Prize to László Lovász of Eötvös Loránd University in Budapest, Hungary and Avi Wigderson of the Institute for Advanced Study, Princeton, USA, “for their foundational contributions to theoretical computer science and discrete mathematics, and their leading role in shaping them into central fields of modern mathematics.” Widely hailed as the mathematical equivalent of the Nobel Prize, this year’s Abel goes to two trailblazers for their revolutionary contributions to the mathematical foundations of computing and information systems. The citation of the award can be found here.

László Lovász

Lovász bagged three gold medals at the International Mathematical Olympiad, from 1964 to 1966, and received his Candidate of Sciences (Ph.D. equivalent) degree in 1970 at the Hungarian Academy of Sciences advised by Tibor Gallai. Lovász was initially based in Hungary, at Eötvös Loránd University and József Attila University, and in 1993 he was appointed William K Lanman Professor of Computer Science and Mathematics at Yale University. In 1999, he had a brief sojourn as a Senior Researcher at Microsoft, before returning in 2006 to Eötvös Loránd University, where he is currently a professor. A product of the legendary Budapest School of Mathematics, László represents the long line of distinguished Hungarian mathematicians right from János Bolyai and the Riesz brothers, Frigyes and Marcel; to Gábor Szegő; Alfréd Rényi; John von Neumann; Paul Turán; Paul Erdős; Raoul Bott, Peter Lax, Endre Szemerédi. Early encounters with Erdős influenced Lovász to work in ‘Hungarian-style combinatorics’, essentially concerned with properties of graphs.

In 1972, Lovász proved the weak perfect graph conjecture, utilising the paradigm of expressing discrete structures by systems of linear inequalities. In 1979, he solved a famous and long-standing open problem on Shannon capacity of the pentagon in the field of information theory, introducing quadratic forms to express discrete structures and heralding in the era of semidefinite programming, a central topic in mathematical optimization and revolutionising algorithm design. He also resolved Kneser’s conjecture, borrowing tools from algebraic topology, and played a pioneering role in the development of the geometric methodology of algorithms based on the ellipsoid method, which led to the solution of a major open problem on submodular function minimization. Through the Lovász local lemma, he provided a fundamental tool of probabilistic methods for the analysis of discrete structures and contributed to the PCP characterization of NP and its connection to the hardness of approximation while simultaneously constructing important algorithms like the matroid matching algorithm and the basis reduction algorithm for integer lattices (LLL). The reduction algorithm, commonly known as the LLL (Lenstra–Lenstra–Lovász lattice basis reduction algorithm named after Lovász and the brothers Arjen and Hendrik Lenstra) algorithm, is one of the basic tools of cryptography and is a fundamental algorithm for solving lattice problems. It was famously used by Andrew Odlyzko and Herman te Riele in disproving the Mertens conjecture. Decades later, LLL still has numerous applications in number theory, integer programming, and cryptography.

Avi Wigderson

Wigderson completed his undergraduate studies at the Technion in Israel and went on to receive a Ph.D. in computer science at Princeton under the supervision of Richard Lipton in 1983. He then served as a Visiting Assistant Professor at UC Berkeley, a Visiting Scientist at IBM, and a Fellow at MSRI in Berkeley before joining the Hebrew University as a faculty member in 1986, and since 1999, Wigderson has been the Herbert H. Maass Professor in the School of Mathematics at the Institute for Advanced Study, Princeton. Wigderson has made profoundly fundamental contributions to the foundations of computing in areas like randomised computation, cryptography, and computational complexity.

In a landmark series of results, Wigderson proved under certain computational assumptions that every probabilistic polynomial time algorithm can be fully derandomized., i.e., randomness is not necessary for polynomial-time computation, providing strong evidence for P = BPP. In cryptography, Wigderson showed how one could compute any function securely in the presence of dishonest parties; showcased the existence of zero-knowledge proofs, proofs that yield nothing but their validity; and helped define the model of multiprover interactive proofs that eventually led to the celebrated PCP theorem. Wigderson also provided lower bounds on the efficiency of communication protocols, circuits, and formal proof systems. and gave the first efficient combinatorial constructions of expander graphs (the zig-zag graph product), an important class of highly connected sparse graphs, and subsequently inspired many important results.

Conclusion

Wigderson and Lovász have additionally, and very successfully, donned hats of terrific mentors, able administrators, wonderful expositors, and inspirational leaders. Their fantastic books and fascinating lectures have stimulated mathematical research around the world. Wigderson has trained many generations of theoretical computer scientists at the Institute for Advanced Study whilst for many, Lovász’s Combinatorial Problems and Exercises is a necessary and serendipitous initiation to the world of combinatorics. Lovász was President of the International Mathematical Union from 2007 to 2010, and also headed the Hungarian Academy of Sciences from 2014 to 2020, strongly pushing back against the farcical policies of the far right government.

Wigderson and Lovász’s phenomenal work has found significantly incredible real-world applications. Brin and Page relied on Lovász’s spectral graph theory results to develop PageRank, the backbone of Google Search, whilst Wigderson’s work on zero knowledge proofs underpins the technology behind blockchains and cryptocurrencies. Apart from applications to devising stronger cryptographic systems, their work catalyses advances in neuroscience, quantum physics, statistical mechanics, economics, social sciences, and medicine, displaying the ubiquity of fundamental basic science research in driving modern day technological advances. Wigderson and Lovász not only introduced visionary and powerful concepts but also solved formidable problems and their work is characterised by extraordinary depth, technical power, creative synthesis of ideas and methods from vast swathes of mathematics and computer science. Through their monumental contributions, effortlessly transcending boundaries between pure and applied mathematics, the two giants of modern day science and mathematics have resoundingly demonstrated that theoretical computer science is fundamentally and intrinsically a part of mathematics.

Some words on Theoretical Computer Science and Discrete Mathematics

Theoretical Computer Science is the study of the power and limitations of computing. It contains two complementary sub-disciplines: algorithm design, which develops efficient methods for a multitude of computational problems; and computational complexity, which shows inherent limitations on the efficiency of algorithms. The notion of polynomial-time algorithms put forward in the 1960s by Alan Cobham, Jack Edmonds, and others, and the famous P≠NP conjecture of Stephen Cook, Leonid Levin, and Richard Karp had a very strong impact on the field.

Discrete mathematics is the study of structures such as graphs, sequences, permutations, and geometric configurations. The combinatorics of discrete structures is also a major component of many areas of pure mathematics, including number theory, probability, algebra, geometry, and analysis. Combinatorics is the study of patterns and graph theory is the study of connections such as in a network. Both come under the umbrella of ‘discrete’ maths, since the objects of study have distinct values, rather than varying smoothly like a point moving along a curve. The mathematics of such structures forms the foundation of theoretical computer science and information theory. For instance, communication networks such as the internet can be described and analyzed using the tools of graph theory, and the design of efficient computational algorithms relies crucially on insights from discrete mathematics.

Ramanujan Graphs and the Matrix Completion Problem

Bishal Deb — Sat, 20 Mar 2021 14:05:36 GMT

Prof. Mathukumalli Vidyasagar FRS, SERB National Science Chair and Distinguished Professor of the Indian Institute of Technology Hyderabad gave a talk on 20th March, 2021 on “Ramanujan Graphs and the Matrix Completion Problem.” The slides of the talk can be found here.

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Gonit Sora

Analysis of Random Forests in predicting the number of 6s hit in IPL

Analyzing the Brachistochrone Problem for the Conic Sections and its real life applications

Analyzing the Brachistochrone Problem for the Conic Sections and its real lfie applications

Abstract

Contents

1 Introduction

2 Generalized integral time formula for any curve and descent for a straight line

3 Time taken for the descent by various conic sections

3.1 Parabola

3.2 Circle

3.3 Ellipse

3.4 Hyperbola

4 Conclusion

References

The Fourier Transform: Bridging Theory and Applications in Signal Processing, Music Synthesis, and Climate Analytics

An Incredibly Short Tale About Human Imagination

We are changing!

Interview with Dr. Cyriac Abby Philips on Evidence Based Medicine

Introduction to Machine Learning

What is Machine Learning?

Major approaches in Machine Learning

Supervised Learning

Methods of supervised learning – Classification and Regression

Unsupervised Learning

Methods of Unsupervised Learning

Semi-supervised Learning

Reinforcement Learning

Mathematics and Machine Learning

Conclusion

2021 Nobel Prize in Physics

Nobel Prize in Medicine 2021

Emmy Noether faced sexism and Nazism – 100 years later her contributions to ring theory still influence modern math

A rare woman in mathematics

A revolution in ring theory

Escape from Nazi Germany

Mathematics of the Football

What is the Riemann Hypothesis, and why does it matter?

ACM Turing Award 2020

Laszlo Lovasz and Avi Wigderson share the 2021 Abel Prize

László Lovász

Avi Wigderson

Conclusion

Some words on Theoretical Computer Science and Discrete Mathematics

Ramanujan Graphs and the Matrix Completion Problem