6 Math Foundations to Start Learning Machine Learning

As a Data Scientist, machine learning is our arsenal to do our job. I am pretty sure in this modern times, everyone who is employed as a Data Scientist would use machine learning to analyze their data to produce valuable patterns. Although, why we need to learn math for machine learning? There is some argument I could give, this includes:

  • Math helps you select the correct machine learning algorithm. Understanding math gives you insight into how the model works, including choosing the right model parameter and the validation strategies.
  • Estimating how confident we are with the model result by producing the right confidence interval and uncertainty measurements needs an understanding of math.
  • The right model would consider many aspects such as metrics, training time, model complexity, number of parameters, and number of features which need math to understand all of these aspects.
  • You could develop a customized model that fits your own problem by knowing the machine learning model’s math.

The main problem is what math subject you need to understand machine learning? Math is a vast field, after all. That is why in this article, I want to outline the math subject you need for machine learning and a few important point to starting learning those subjects.

Machine Learning Math

We could learn many topics from the math subject, but if we want to focus on the math used in machine learning, we need to specify it. In this case, I like to use the necessary math references explained in the Machine Learning Math book by M. P. Deisenroth, A. A. Faisal, and C. S. Ong, 2021.

In their book, there are math foundations that are important for Machine Learning. The math subject is:

Image created by Author

Six math subjects become the foundation for machine learning. Each subject is intertwined to develop our machine learning model and reach the “best” model for generalizing the dataset.

Let’s dive deeper for each subject to know what they are.

Linear Algebra

What is Linear Algebra? This is a branch of mathematic that concerns the study of the vectors and certain rules to manipulate the vector. When we are formalizing intuitive concepts, the common approach is to construct a set of objects (symbols) and a set of rules to manipulate these objects. This is what we knew as algebra.

If we talk about Linear Algebra in machine learning, it is defined as the part of mathematics that uses vector space and matrices to represent linear equations.

When talking about vectors, people might flashback to their high school study regarding the vector with direction, just like the image below.

Geometric Vector (Image by Author)

This is a vector, but not the kind of vector discussed in the Linear Algebra for Machine Learning. Instead, it would be this image below we would talk about.

Vector 4×1 Matrix (Image by Author)

What we had above is also a Vector, but another kind of vector. You might be familiar with matrix form (the image below). The vector is a matrix with only 1 column, which is known as a column vector. In other words, we can think of a matrix as a group of column vectors or row vectors. In summary, vectors are special objects that can be added together and multiplied by scalars to produce another object of the same kind. We could have various objects called vectors.

Matrix (Image by Author)

Linear algebra itself s a systematic representation of data that computers can understand, and all the operations in linear algebra are systematic rules. That is why in modern time machine learning, Linear algebra is an important study.

An example of how linear algebra is used is in the linear equation. Linear algebra is a tool used in the Linear Equation because so many problems could be presented systematically in a Linear way. The typical Linear equation is presented in the form below.

Linear Equation (Image by Author)

To solve the linear equation problem above, we use Linear Algebra to present the linear equation in a systematical representation. This way, we could use the matrix characterization to look for the most optimal solution.

Linear Equation in Matrix Representation (Image by Author)

To summary the Linear Algebra subject, there are three terms you might want to learn more as a starting point within this subject:

  • Vector
  • Matrix
  • Linear Equation

Analytic Geometry (Coordinate Geometry)

Analytic geometry is a study in which we learn the data (point) position using an ordered pair of coordinates. This study is concerned with defining and representing geometrical shapes numerically and extracting numerical information from the shapes numerical definitions and representations. We project the data into the plane in a simpler term, and we receive numerical information from there.

Cartesian Coordinate (Image by Author)

Above is an example of how we acquired information from the data point by projecting the dataset into the plane. How we acquire the information from this representation is the heart of Analytical Geometry. To help you start learning this subject, here are some important terms you might need.

  • Distance Function

A distance function is a function that provides numerical information for the distance between the elements of a set. If the distance is zero, then elements are equivalent. Else, they are different from each other.

An example of the distance function is Euclidean Distance which calculates the linear distance between two data points.

Euclidean Distance Equation (Image by Author)
  • Inner Product

The inner product is a concept that introduces intuitive geometrical concepts, such as the length of a vector and the angle or distance between two vectors. It is often denoted as ⟨x,y⟩ (or occasionally (x,y) or ⟨x|y⟩).

Matrix Decomposition

Matrix Decomposition is a study that concerning the way to reducing a matrix into its constituent parts. Matrix Decomposition aims to simplify more complex matrix operations on the decomposed matrix rather than on its original matrix.

A common analogy for matrix decomposition is like factoring numbers, such as factoring 8 into 2 x 4. This is why matrix decomposition is synonymical to matrix factorization. There are many ways to decompose a matrix, so there is a range of different matrix decomposition techniques. An example is the LU Decomposition in the image below.

LU Decomposition (Image by Author)

Vector Calculus

Calculus is a mathematical study that concern with continuous change, which mainly consists of functions and limits. Vector calculus itself is concerned with the differentiation and integration of the vector fields. Vector Calculus is often called multivariate calculus, although it has a slightly different study case. Multivariate calculus deals with calculus application functions of the multiple independent variables.

There are a few important terms I feel people need to know when starting learning the Vector Calculus, they are:

  • Derivative and Differentiation

The derivative is a function of real numbers that measure the change of the function value (output value) concerning a change in its argument (input value). Differentiation is the action of computing a derivative.

Derivative Equation (Image by Author)
  • Partial Derivative

The partial derivative is a derivative function where several variables are calculated within the derivative function with respect to one of those variables could be varied, and the other variable are held constant (as opposed to the total derivative, in which all variables are allowed to vary).

  • Gradient

The gradient is a word related to the derivative or the rate of change of a function; you might consider that gradient is a fancy word for derivative. The term gradient is typically used for functions with several inputs and a single output (scalar). The gradient has a direction to move from their current location, e.g., up, down, right, left.

Probability and Distribution

Probability is a study of uncertainty (loosely terms). The probability here can be thought of as a time where the event occurs or the degree of belief about an event’s occurrence. The probability distribution is a function that measures the probability of a particular outcome (or probability set of outcomes) that would occur associated with the random variable. The common probability distribution function is shown in the image below.

Normal Distribution Probability Function (Image by Author)

Probability theory and statistics are often associated with a similar thing, but they concern different aspects of uncertainty:

•In math, we define probability as a model of some process where random variables capture the underlying uncertainty, and we use the rules of probability to summarize what happens.

•In statistics, we try to figure out the underlying process observe of something that has happened and tries to explain the observations.

When we talk about machine learning, it is close to statistics because its goal is to construct a model that adequately represents the process that generated the data.

Optimization

In the learning objective, training a machine learning model is all about finding a good set of parameters. What we consider “good” is determined by the objective function or the probabilistic models. This is what optimization algorithms are for; given an objective function, we try to find the best value.

Commonly, objective functions in machine learning are trying to minimize the function. It means the best value is the minimum value. Intuitively, if we try to find the best value, it would like finding the valleys of the objective function where the gradients point us uphill. That is why we want to move downhill (opposite to the gradient) and hope to find the lowest (deepest) point. This is the concept of gradient descent.

Gradient Descent (Image by Author)

There are few terms as a starting point when learning optimization. They are:

  • Local Minima and Global Minima

The point at which a function best values takes the minimum value is called the global minima. However, when the goal is to minimize the function and solved it using optimization algorithms such as gradient descent, the function could have a minimum value at different points. Those several points which appear to be minima but are not the point where the function actually takes the minimum value are called local minima.

Local and Global Minima (Image by Author)
  • Unconstrained Optimization and Constrained Optimization

Unconstrained Optimization is an optimization function where we find a minimum of a function under the assumption that the parameters can take any possible value (no parameter limitation). Constrained Optimization simply limits the possible value by introducing a set of constraints.

Gradient descent is an Unconstrained optimization if there is no parameter limitation. If we set some limit, for example, x > 1, it is an unconstrained optimization.

Conclusion

Machine Learning is an everyday tool that Data scientists use to obtain the valuable pattern we need. Learning the math behind machine learning could provide you an edge in your work. There are many math subjects out there, but there are 6 subjects that matter the most when we are starting learning machine learning math, and that is:

  • Linear Algebra
  • Analytic Geometry
  • Matrix Decomposition
  • Vector Calculus
  • Probability and Distribution
  • Optimization

If you start learning math for machine learning, you could read my other article to avoid the study pitfall. I also provide the math material you might want to check out in that article.

 

By: Cornellius Yudha Wijaya

Source: 6 Math Foundations to Start Learning Machine Learning | by Cornellius Yudha Wijaya | Towards Data Science

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Critics:

Machine learning (ML) is the study of computer algorithms that improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data, known as “training data“, in order to make predictions or decisions without being explicitly programmed to do so. Machine learning algorithms are used in a wide variety of applications, such as in medicine, email filtering, speech recognition, and computer vision, where it is difficult or unfeasible to develop conventional algorithms to perform the needed tasks.

A subset of machine learning is closely related to computational statistics, which focuses on making predictions using computers; but not all machine learning is statistical learning. The study of mathematical optimization delivers methods, theory and application domains to the field of machine learning. Data mining is a related field of study, focusing on exploratory data analysis through unsupervised learning. In its application across business problems, machine learning is also referred to as predictive analytics.

Machine learning approaches are traditionally divided into three broad categories, depending on the nature of the “signal” or “feedback” available to the learning system:

  • Supervised learning: The computer is presented with example inputs and their desired outputs, given by a “teacher”, and the goal is to learn a general rule that maps inputs to outputs.
  • Unsupervised learning: No labels are given to the learning algorithm, leaving it on its own to find structure in its input. Unsupervised learning can be a goal in itself (discovering hidden patterns in data) or a means towards an end (feature learning).
  • Reinforcement learning: A computer program interacts with a dynamic environment in which it must perform a certain goal (such as driving a vehicle or playing a game against an opponent). As it navigates its problem space, the program is provided feedback that’s analogous to rewards, which it tries to maximize.

References

Physicists Debate Hawking’s Idea That the Universe Had No Beginning

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In 1981, many of the world’s leading cosmologists gathered at the Pontifical Academy of Sciences, a vestige of the coupled lineages of science and theology located in an elegant villa in the gardens of the Vatican. Stephen Hawking chose the august setting to present what he would later regard as his most important idea: a proposal about how the universe could have arisen from nothing.

Before Hawking’s talk, all cosmological origin stories, scientific or theological, had invited the rejoinder, “What happened before that?” The Big Bang theory, for instance — pioneered 50 years before Hawking’s lecture by the Belgian physicist and Catholic priest Georges Lemaître, who later served as president of the Vatican’s academy of sciences — rewinds the expansion of the universe back to a hot, dense bundle of energy. But where did the initial energy come from?

The Big Bang theory had other problems. Physicists understood that an expanding bundle of energy would grow into a crumpled mess rather than the huge, smooth cosmos that modern astronomers observe. In 1980, the year before Hawking’s talk, the cosmologist Alan Guth realized that the Big Bang’s problems could be fixed with an add-on: an initial, exponential growth spurt known as cosmic inflation, which would have rendered the universe huge, smooth and flat before gravity had a chance to wreck it. Inflation quickly became the leading theory of our cosmic origins. Yet the issue of initial conditions remained: What was the source of the minuscule patch that allegedly ballooned into our cosmos, and of the potential energy that inflated it?

Hawking, in his brilliance, saw a way to end the interminable groping backward in time: He proposed that there’s no end, or beginning, at all. According to the record of the Vatican conference, the Cambridge physicist, then 39 and still able to speak with his own voice, told the crowd, “There ought to be something very special about the boundary conditions of the universe, and what can be more special than the condition that there is no boundary?”

The “no-boundary proposal,” which Hawking and his frequent collaborator, James Hartle, fully formulated in a 1983 paper, envisions the cosmos having the shape of a shuttlecock. Just as a shuttlecock has a diameter of zero at its bottommost point and gradually widens on the way up, the universe, according to the no-boundary proposal, smoothly expanded from a point of zero size. Hartle and Hawking derived a formula describing the whole shuttlecock — the so-called “wave function of the universe” that encompasses the entire past, present and future at once — making moot all contemplation of seeds of creation, a creator, or any transition from a time before.

“Asking what came before the Big Bang is meaningless, according to the no-boundary proposal, because there is no notion of time available to refer to,” Hawking said in another lecture at the Pontifical Academy in 2016, a year and a half before his death. “It would be like asking what lies south of the South Pole.”

Hartle and Hawking’s proposal radically reconceptualized time. Each moment in the universe becomes a cross-section of the shuttlecock; while we perceive the universe as expanding and evolving from one moment to the next, time really consists of correlations between the universe’s size in each cross-section and other properties — particularly its entropy, or disorder. Entropy increases from the cork to the feathers, aiming an emergent arrow of time. Near the shuttlecock’s rounded-off bottom, though, the correlations are less reliable; time ceases to exist and is replaced by pure space. As Hartle, now 79 and a professor at the University of California, Santa Barbara, explained it by phone recently, “We didn’t have birds in the very early universe; we have birds later on. … We didn’t have time in the early universe, but we have time later on.”

The no-boundary proposal has fascinated and inspired physicists for nearly four decades. “It’s a stunningly beautiful and provocative idea,” said Neil Turok, a cosmologist at the Perimeter Institute for Theoretical Physics in Waterloo, Canada, and a former collaborator of Hawking’s. The proposal represented a first guess at the quantum description of the cosmos — the wave function of the universe. Soon an entire field, quantum cosmology, sprang up as researchers devised alternative ideas about how the universe could have come from nothing, analyzed the theories’ various predictions and ways to test them, and interpreted their philosophical meaning. The no-boundary wave function, according to Hartle, “was in some ways the simplest possible proposal for that.”

But two years ago, a paper by Turok, Job Feldbrugge of the Perimeter Institute, and Jean-Luc Lehners of the Max Planck Institute for Gravitational Physics in Germany called the Hartle-Hawking proposal into question. The proposal is, of course, only viable if a universe that curves out of a dimensionless point in the way Hartle and Hawking imagined naturally grows into a universe like ours. Hawking and Hartle argued that indeed it would — that universes with no boundaries will tend to be huge, breathtakingly smooth, impressively flat, and expanding, just like the actual cosmos. “The trouble with Stephen and Jim’s approach is it was ambiguous,” Turok said — “deeply ambiguous.”

In their 2017 paper, published in Physical Review Letters, Turok and his co-authors approached Hartle and Hawking’s no-boundary proposal with new mathematical techniques that, in their view, make its predictions much more concrete than before. “We discovered that it just failed miserably,” Turok said. “It was just not possible quantum mechanically for a universe to start in the way they imagined.” The trio checked their math and queried their underlying assumptions before going public, but “unfortunately,” Turok said, “it just seemed to be inescapable that the Hartle-Hawking proposal was a disaster.”

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The paper ignited a controversy. Other experts mounted a vigorous defense of the no-boundary idea and a rebuttal of Turok and colleagues’ reasoning. “We disagree with his technical arguments,” said Thomas Hertog, a physicist at the Catholic University of Leuven in Belgium who closely collaborated with Hawking for the last 20 years of the latter’s life. “But more fundamentally, we disagree also with his definition, his framework, his choice of principles. And that’s the more interesting discussion.”

After two years of sparring, the groups have traced their technical disagreement to differing beliefs about how nature works. The heated — yet friendly — debate has helped firm up the idea that most tickled Hawking’s fancy. Even critics of his and Hartle’s specific formula, including Turok and Lehners, are crafting competing quantum-cosmological models that try to avoid the alleged pitfalls of the original while maintaining its boundless allure.

Garden of Cosmic Delights

Hartle and Hawking saw a lot of each other from the 1970s on, typically when they met in Cambridge for long periods of collaboration. The duo’s theoretical investigations of black holes and the mysterious singularities at their centers had turned them on to the question of our cosmic origin.

In 1915, Albert Einstein discovered that concentrations of matter or energy warp the fabric of space-time, causing gravity. In the 1960s, Hawking and the Oxford University physicist Roger Penrose proved that when space-time bends steeply enough, such as inside a black hole or perhaps during the Big Bang, it inevitably collapses, curving infinitely steeply toward a singularity, where Einstein’s equations break down and a new, quantum theory of gravity is needed. The Penrose-Hawking “singularity theorems” meant there was no way for space-time to begin smoothly, undramatically at a point.

GRAPHIC: "In the ‘Beginning’"
5W Infographics for Quanta Magazine

Hawking and Hartle were thus led to ponder the possibility that the universe began as pure space, rather than dynamical space-time. And this led them to the shuttlecock geometry. They defined the no-boundary wave function describing such a universe using an approach invented by Hawking’s hero, the physicist Richard Feynman. In the 1940s, Feynman devised a scheme for calculating the most likely outcomes of quantum mechanical events. To predict, say, the likeliest outcomes of a particle collision, Feynman found that you could sum up all possible paths that the colliding particles could take, weighting straightforward paths more than convoluted ones in the sum. Calculating this “path integral” gives you the wave function: a probability distribution indicating the different possible states of the particles after the collision.

Likewise, Hartle and Hawking expressed the wave function of the universe — which describes its likely states — as the sum of all possible ways that it might have smoothly expanded from a point. The hope was that the sum of all possible “expansion histories,” smooth-bottomed universes of all different shapes and sizes, would yield a wave function that gives a high probability to a huge, smooth, flat universe like ours. If the weighted sum of all possible expansion histories yields some other kind of universe as the likeliest outcome, the no-boundary proposal fails.

The problem is that the path integral over all possible expansion histories is far too complicated to calculate exactly. Countless different shapes and sizes of universes are possible, and each can be a messy affair. “Murray Gell-Mann used to ask me,” Hartle said, referring to the late Nobel Prize-winning physicist, “if you know the wave function of the universe, why aren’t you rich?” Of course, to actually solve for the wave function using Feynman’s method, Hartle and Hawking had to drastically simplify the situation, ignoring even the specific particles that populate our world (which meant their formula was nowhere close to being able to predict the stock market). They considered the path integral over all possible toy universes in “minisuperspace,” defined as the set of all universes with a single energy field coursing through them: the energy that powered cosmic inflation. (In Hartle and Hawking’s shuttlecock picture, that initial period of ballooning corresponds to the rapid increase in diameter near the bottom of the cork.)

Even the minisuperspace calculation is hard to solve exactly, but physicists know there are two possible expansion histories that potentially dominate the calculation. These rival universe shapes anchor the two sides of the current debate.

The rival solutions are the two “classical” expansion histories that a universe can have. Following an initial spurt of cosmic inflation from size zero, these universes steadily expand according to Einstein’s theory of gravity and space-time. Weirder expansion histories, like football-shaped universes or caterpillar-like ones, mostly cancel out in the quantum calculation.

One of the two classical solutions resembles our universe. On large scales, it’s smooth and randomly dappled with energy, due to quantum fluctuations during inflation. As in the real universe, density differences between regions form a bell curve around zero. If this possible solution does indeed dominate the wave function for minisuperspace, it becomes plausible to imagine that a far more detailed and exact version of the no-boundary wave function might serve as a viable cosmological model of the real universe.

The other potentially dominant universe shape is nothing like reality. As it widens, the energy infusing it varies more and more extremely, creating enormous density differences from one place to the next that gravity steadily worsens. Density variations form an inverted bell curve, where differences between regions approach not zero, but infinity. If this is the dominant term in the no-boundary wave function for minisuperspace, then the Hartle-Hawking proposal would seem to be wrong.

The two dominant expansion histories present a choice in how the path integral should be done. If the dominant histories are two locations on a map, megacities in the realm of all possible quantum mechanical universes, the question is which path we should take through the terrain. Which dominant expansion history, and there can only be one, should our “contour of integration” pick up? Researchers have forked down different paths.

In their 2017 paper, Turok, Feldbrugge and Lehners took a path through the garden of possible expansion histories that led to the second dominant solution. In their view, the only sensible contour is one that scans through real values (as opposed to imaginary values, which involve the square roots of negative numbers) for a variable called “lapse.” Lapse is essentially the height of each possible shuttlecock universe — the distance it takes to reach a certain diameter. Lacking a causal element, lapse is not quite our usual notion of time. Yet Turok and colleagues argue partly on the grounds of causality that only real values of lapse make physical sense. And summing over universes with real values of lapse leads to the wildly fluctuating, physically nonsensical solution.

“People place huge faith in Stephen’s intuition,” Turok said by phone. “For good reason — I mean, he probably had the best intuition of anyone on these topics. But he wasn’t always right.”

Imaginary Universes

Jonathan Halliwell, a physicist at Imperial College London, has studied the no-boundary proposal since he was Hawking’s student in the 1980s. He and Hartle analyzed the issue of the contour of integration in 1990. In their view, as well as Hertog’s, and apparently Hawking’s, the contour is not fundamental, but rather a mathematical tool that can be placed to greatest advantage. It’s similar to how the trajectory of a planet around the sun can be expressed mathematically as a series of angles, as a series of times, or in terms of any of several other convenient parameters. “You can do that parameterization in many different ways, but none of them are any more physical than another one,” Halliwell said.

He and his colleagues argue that, in the minisuperspace case, only contours that pick up the good expansion history make sense. Quantum mechanics requires probabilities to add to 1, or be “normalizable,” but the wildly fluctuating universe that Turok’s team landed on is not. That solution is nonsensical, plagued by infinities and disallowed by quantum laws — obvious signs, according to no-boundary’s defenders, to walk the other way.

It’s true that contours passing through the good solution sum up possible universes with imaginary values for their lapse variables. But apart from Turok and company, few people think that’s a problem. Imaginary numbers pervade quantum mechanics. To team Hartle-Hawking, the critics are invoking a false notion of causality in demanding that lapse be real. “That’s a principle which is not written in the stars, and which we profoundly disagree with,” Hertog said.

According to Hertog, Hawking seldom mentioned the path integral formulation of the no-boundary wave function in his later years, partly because of the ambiguity around the choice of contour. He regarded the normalizable expansion history, which the path integral had merely helped uncover, as the solution to a more fundamental equation about the universe posed in the 1960s by the physicists John Wheeler and Bryce DeWitt. Wheeler and DeWitt — after mulling over the issue during a layover at Raleigh-Durham International — argued that the wave function of the universe, whatever it is, cannot depend on time, since there is no external clock by which to measure it. And thus the amount of energy in the universe, when you add up the positive and negative contributions of matter and gravity, must stay at zero forever. The no-boundary wave function satisfies the Wheeler-DeWitt equation for minisuperspace.  

In the final years of his life, to better understand the wave function more generally, Hawking and his collaborators started applying holography — a blockbuster new approach that treats space-time as a hologram. Hawking sought a holographic description of a shuttlecock-shaped universe, in which the geometry of the entire past would project off of the present.

That effort is continuing in Hawking’s absence. But Turok sees this shift in emphasis as changing the rules. In backing away from the path integral formulation, he says, proponents of the no-boundary idea have made it ill-defined. What they’re studying is no longer Hartle-Hawking, in his opinion — though Hartle himself disagrees.

For the past year, Turok and his Perimeter Institute colleagues Latham Boyle and Kieran Finn have been developing a new cosmological model that has much in common with the no-boundary proposal. But instead of one shuttlecock, it envisions two, arranged cork to cork in a sort of hourglass figure with time flowing in both directions. While the model is not yet developed enough to make predictions, its charm lies in the way its lobes realize CPT symmetry, a seemingly fundamental mirror in nature that simultaneously reflects matter and antimatter, left and right, and forward and backward in time. One disadvantage is that the universe’s mirror-image lobes meet at a singularity, a pinch in space-time that requires the unknown quantum theory of gravity to understand. Boyle, Finn and Turok take a stab at the singularity, but such an attempt is inherently speculative.

There has also been a revival of interest in the “tunneling proposal,” an alternative way that the universe might have arisen from nothing, conceived in the ’80s independently by the Russian-American cosmologists Alexander Vilenkin and Andrei Linde. The proposal, which differs from the no-boundary wave function primarily by way of a minus sign, casts the birth of the universe as a quantum mechanical “tunneling” event, similar to when a particle pops up beyond a barrier in a quantum mechanical experiment.

Questions abound about how the various proposals intersect with anthropic reasoning and the infamous multiverse idea. The no-boundary wave function, for instance, favors empty universes, whereas significant matter and energy are needed to power hugeness and complexity. Hawking argued that the vast spread of possible universes permitted by the wave function must all be realized in some larger multiverse, within which only complex universes like ours will have inhabitants capable of making observations. (The recent debate concerns whether these complex, habitable universes will be smooth or wildly fluctuating.) An advantage of the tunneling proposal is that it favors matter- and energy-filled universes like ours without resorting to anthropic reasoning — though universes that tunnel into existence may have other problems.

No matter how things go, perhaps we’ll be left with some essence of the picture Hawking first painted at the Pontifical Academy of Sciences 38 years ago. Or perhaps, instead of a South Pole-like non-beginning, the universe emerged from a singularity after all, demanding a different kind of wave function altogether. Either way, the pursuit will continue. “If we are talking about a quantum mechanical theory, what else is there to find other than the wave function?” asked Juan Maldacena, an eminent theoretical physicist at the Institute for Advanced Study in Princeton, New Jersey, who has mostly stayed out of the recent fray. The question of the wave function of the universe “is the right kind of question to ask,” said Maldacena, who, incidentally, is a member of the Pontifical Academy. “Whether we are finding the right wave function, or how we should think about the wave function — it’s less clear.”

BY: Natalie Wolchover

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The ‘Nobel Prize of Math’ Has Been Won By A Woman For The First Time Ever

Greetings with some good news for the women’s world. Just recently, one of the most prestigious mathematics prizes in the world – The Abel Prize was awarded to a woman for the first time ever. Yes! Karen Uhlenbeck is a mathematician and a professor at the University of Texas and is now the first woman to win this prize in mathematics. You go Karen!

The award, which is modeled by the Nobel Prize, is awarded by the king of Norway to honor mathematicians who have made an influence in their field including a cash prize of around $700,000. The award to Karen cites for “the fundamental impact of her work on analysis, geometry and mathematical physics.” This award exists since 2003 but has only been won by men since.

Among her colleagues, Dr. Uhlenbeck is renowned for her work in geometric partial differential equations as well as integrable systems and gauge theory. One of her most famous contributions were her theories of predictive mathematics and in pioneering the field of geometric analysis.

Sun-Yung Alice Chang, a mathematician at Princeton University who was in the prize committee says about her: “She did things nobody thought about doing, and after she did, she laid the foundations of a branch of mathematics.”

Source: The ‘Nobel Prize of Math’ Has Been Won By A Woman For The First Time Ever

Topic: Morrey Spaces and Regularity for Yang-Mills Higgs Equations
Speaker: Karen Uhlenbeck
Affiliation: School of Mathematics
Date: December 7, 2018

The Microseconds That Can Rule Out Relative Time! — Lucid Being

The Microseconds That Can Rule Out Relative Time! According to Albert Einstein’s Theory Of Special Relativity, your time and my time are different, subject to and conditional to the question of your speed of movement and my speed of movement. The speed in which we are moving toward each other or the speed in which […]

via The Microseconds That Can Rule Out Relative Time! — Lucid Being

Is Anti-Gravity Real? Science Is About To Find Out – Ethan Siegel

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One of the most astonishing facts about science is how universally applicable the laws of nature are. Every particle obeys the same rules, experiences the same forces, and sees the same fundamental constants, no matter where or when they exist. Gravitationally, every single entity in the Universe experiences, depending on how you look at it, either the same gravitational acceleration or the same curvature of spacetime, no matter what properties it possesses. At least, that’s what things are like in theory. In practice, some things are notoriously difficult to measure……..

Read more: https://www.forbes.com/sites/startswithabang/2018/10/26/is-anti-gravity-real-science-is-about-to-find-out/#4de36b849a79

 

 

 

 

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Class Pad: Free digital scratch paper for the math classroom – via Kelly Tenkely | iGeneration – 21st Century Education (Pedagogy & Digital Innovation)

Class Pad: Free digital scratch paper for the math classroom – via Kelly Tenkely on iGeneration – 21st Century Education (Pedagogy & Digital Innovation) curated by Tom D’Amico (@TDOttawa)

Source: Class Pad: Free digital scratch paper for the math classroom – via Kelly Tenkely | iGeneration – 21st Century Education (Pedagogy & Digital Innovation)

Why Mathematicians Can’t Find the Hay in a Haystack – Vladyslav Danilin

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The first time I heard a mathematician use the phrase, I was sure he’d misspoken. We were on the phone, talking about the search for shapes with certain properties, and he said, “It’s like looking for hay in a haystack.” “Don’t you mean a needle?” I almost interjected. Then he said it again. In mathematics, it turns out, conventional modes of thought sometimes get turned on their head. The mathematician I was speaking with, Dave Jensen of the University of Kentucky, really did mean “hay in a haystack.” By it, he was expressing a strange fact about mathematical research: Sometimes the most common things are the hardest to find…….

Read more: https://www.quantamagazine.org/why-mathematicians-cant-find-the-hay-in-a-haystack-20180917/

 

 

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