Mathematical Modeling

A new math book each blog-post 

There are quite a few books on mathematical modeling available out there, but I want to literally and figuratively focus  on a single one, which is Mathematical Modeling by Mark M. Meerschaert.

First, a number of details about the author of the book. Mark Meerschaert is  a University Distinguished Professor in the Department of Statistics and Probability at Michigan State University. He authored a number of books among them the Mathematical Modeling.

What is special about the book?

I have a third edition of the book and I want to provide some thoughts about it. Personally, I like books that provide detailed explanations and ample of examples accompanying the theoretical parts of the book. In my opinion, author’s own view on the subject phrased in his own words, instead of strict adherence to formal definitions is a valuable aid in comprehending mathematical theory.

As for the content of the book, it is divided in three parts which reflects the fact that most of the mathematical models fall into three types 

  • Optimization Models
  • Dynamic Models 
  • Probability Models

Each chapter in the book has detailed examples and quite a few exercises for the reader to tackle. What is also nice that the book is quite practical and have examples from various fields of science and engineering.

References

mark

Math books Applied for Good

Math books and more books on math

Following the path of applied mathematics and popular science with math inclination I want to bring to your attention a couple of books that some of you may find helpful if not insightful.

Oliver Heaviside’s Maxwell’s Equations

Actually, I would rather start from a book which is an amalgam of history and mathematical physics in one and it’s a book about the self-taught mathematical physicist Oliver Heaviside who brought to you the so called Four Maxwell’s equations.

equations

book_heaviside

The book is Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age written by Paul J. Nahin an emeritus prof. of Electrical Engineering in University Of New Hampshire which we’ll return to later in the post.  What is interesting about the books is that it has a right amount of math for readers who are interested not only to know who Oliver Heaviside was,  but also what he did as a physicist and engineer.

 

 

 

 

Okay, the books

While reading very interesting Applied Mathematics book by David J. Logan (3rd edition, Ch. 4.4 Green’s Functions, p. 253)

step_function

I was, as always, diverged by the mentioning of the Heaviside Step function in the text that I felt an urgent surge to check a biography of this incredible person and, lo an behold, I was able to find the Paul Nahin’s book mentioned above and also quite interesting and short  article in the Physics Today magazine Oliver Heaviside: A first-rate oddity

David J. Logan

Having mentioned, David Logan I should say that I am reading the 3rd edition of his book, which is available in Scribd if you have a membership there, and even for free for 30 days trial period. It is always possible to buy the 4th edition, but the price is, quite frankly, astronomical.

logan

Applied Mathematics 4th edition by David J. Logan. What I like about this book is the detailed examples that help you understand the content of the book better, but even more I like the way David Logan explains the physical rational behind the differential equations. It helps very much to know how and why this or that math technique is applied in practice. In addition, another applied mathematician Mark H. Holmes book’s is also mentioned by David Logan which you also may find useful.

 

 

 

Paul J. Nahin

Now that’s get back to Paul Nahin. It turn’s out he produced a whole series of books on Physics, Mathematics, Electrical Engineering and Computer Science which can be called popular, but actually are an essays full of wonderful applied mathematics. Paul is able to explain things in engaging and easy to understand manner. As people like to say, I wish I had come across his books earlier in my life, but it is what it is and it’s good that I was able to find them. Thanks to the Scribd digital library I was able to glimpsed through all of his books available there and I’d recommend to math inclined readers to check the following books.

simple_physics

 In Praise of Simple Physics: The Science and Mathematics behind Everyday Questions will take you into the physics journey that you could have been missing since your school or collage years. Maybe, you weren’t able  to understand it back then or had no time, but this time it will be different thanks to Paul’s ability to explain physics in an easy to grasp way.

 

 

 

 

And one additional book that I find quite impressing 

crunchung numbers

Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction as all books by Paul J. Nahin this one draws examples from different areas of exact sciences and engineering that will keep you awake at night following the stories and trying to solve the puzzles yourself.

 

 

 

 

Mark H. Holmes

holmes

Remember, I’ve mentioned Mark H. Holmes so he also wrote a couple of books on applied math, and I’d recommend you to check his Introduction to Numerical Methods in Differential Equations which I find also very useful and a helper while reading aforementioned books on applied math. Unlike his Introduction to the Foundations of Applied Mathematics, which I find cryptic due to the lack of detailed examples, Introduction to Numerical Methods has quite a few of them. This makes the book kind of easy to digest.

 

 

 

Last, but not least

To make sense in this whole unfamiliar forest of applied mathematics there is a nice book that has all you need in one place classified and summarized to be your guidance on your quest to master the math and apply it for good. It is

all_of_it

The Princeton Companion To Applied Mathematics.

 

 

 

 

 

 

 

Dare think, keep on going, and be carried forward on wings of math muse.

References

Math is in the air

Start this year the right way

New year’s time is usually a time to make some new year’s resolutions. I won’t do it and instead this year for me at least will be solely focused on applied mathematics. Math topics interest  me for a long time. But I never took it seriously to invest quality time into studying advanced math topics with enough detail. This year will be different. The plan is to start from some quite general books on math that try to approach the topic in an engaging way like Measurement book by Paul Lockhart and slowly transitioning to more technical books for applied mathematics like Elements of Applied Mathematics by Zeldovich and Myskis and  Nonlinear Dynamics and Chaos by Steven H. Strogatz.

A little bit about the books

Why these three books you may wonder? Actually, there are four books I want to focus on. What is so specially about these books is the fact that they do not simply talk about one specific field in math with a very narrow focus on the subject, but like Donald Knuth’s The Art Of Computer Programming volumes approach the subject in a more general way without being fearful to delve in fields of physics, engineering, biology etc.

To summarize the books are:

measurement

Measurement by Paul Lockhart which tries to show math as an engaging activity that resembles arts, such as music, painting where there is a place for creativity,  a joy of new discoveries or a pain of being stuck trying to get a solution.

 

 

 

 

2020-01-01 16_03_56-Elements Of Applied Mathematics _ YA. B. Zeldovich, A. D. Myskis _ Free DownloadElements of Applied Mathematics by Zeldovich and Myskis which is an old book, but it’s still relevant today, at least in many parts of it, as it was back in 1972. As authors themselves put in the foreword of the book

So our advice is: read our book and study it. But even if there is not time enough to make a deep study, then merely read it it like a novel, and it may be just the thing you will need for solving some difficult problem.

 

 

math_arnoldMathematical Understanding of Nature: Essays on Amazing Physical Phenomena and Their Understanding by Mathematicians by Vladimir Arnold. This book is a collection of applied math problems  that were drawn from various fields such as physics, engineering etc.

Note: If you’re capable of reading in Russian then this book is available in full for free here.

 

 

nonlinear-dynamics-and-chaosAnd finally there is the voluminous  Nonlinear Dynamics and Chaos by Steven H. Strogatz that makes me feel better by having hundreds of differential quotations.

Note: The older edition of this book is available for free, for example, here.

A unique feature of the book is its emphasis on applications. These include mechanical vibrations, lasers, biological rhythms, superconducting circuits, insect outbreaks, chemical oscillators, genetic control systems, chaotic waterwheels, and even a technique for using chaos to send secret messages.

 

Now is the best time to start

This paragraph is dedicated to myself and possibly you. Remember that now is the best time to start doing what you wanted, but postponed ad infinitum. So start by taking small steps to big results later. Or at least to having satisfaction from solving some non-trivial tasks and applying a new skill to real life problems.

Thoughts on physics and artificial intelligence on 2019 New Year’s eve

Make the New Year happy, because you can

It seems to me the New Year will be interesting and exciting as it always seems this way on new year’s eve. What makes me think so though is a number of books I read recently. One of the book is a collection of interviews with prominent people in the field that is known as Artificial Intelligence. The other book is about the particle physics being stuck with high hopes in String theory and why it may be a root cause of not seeing no new physics discovered so far in the Large Hadron Collider (LHC) except for Higgs boson.  

The power of the right books

In his book Architects of Intelligence: The truth about AI from the people building it Martin Ford has done something interesting by combining a numbers of interviews, more than a dozen, with people who are focused on Artificial Intelligence progress in various levels. In it you may find Geoff Hinton the founding father of Deep Learning and his colleagues Yoshua Bengio and Yann LeCun who need no special advertising (hint, search in Google). There are also a row of interviews with people like Jeff Dean and Ray Kurzweil from Google Brain that are interesting to read too.

The main point of the book is that those people were asked more or less the same questions, including how they came into field of Artificial Intelligence, what they think about Deep Learning and whether it will alone lead to Artificial General Intelligence. Will the recent advances in machine learning jeopardize jobs and what to do about that. What is interesting to see is that each person interviewed naturally had a different answer to these questions, so it helps to get a balanced view on what is the state of the art of Deep and Machine Learning in 2018.

Things that require new explanations

In her book Lost in Math: How Beauty Leads Physics Astray Sabine Hossenfelder a particle physicist discusses an interesting matter of various biases that affect theoretical physicists that set out to devise a theory that intended to explain laws of physics. For example, String Theory is discussed extensively in the book since this theory though it’s very elegant, beautiful and full of  naturalness completely failed due to the absence of any predictions that the theory envisioned. Indeed, no new particles except for Higgs boson, were found in the Large Hadron Collider and it feels like there is a time to abandon String Theory which isn’t working and check other theories that won’t be plagued with ad hoc assumptions of naturalness and apparent, and very likely deceiving, beauty of the nature. If you are interested why there was found nothing new in the particle physics in recent decades, you may find Sabine’s explanations insightful. And maybe just, maybe you’ll discover that you too like me have biases that affect our perception of the nature. 

 

So make the upcoming year as you wish it to be

Remember that as intelligent creatures we are chanced to possess a capability to set goals and achieve them when we plan and act on plans with an enthusiasm and a perseverance (and Google search).

Happy New Year!

 

 

Language Acquisition. Multidisciplinary approach. Part three.

 

no_frame

Multidisciplinary approach

This post about the research of natural language acquisition will be as short as previous parts. This time I want to describe how the current research that is too linguistically focused may benefit from being open up to other disciplines, such as Machine Learning, computer Science, Neurosciences and Physics.

Currently, the language acquisition research is predominantly done by linguists. In my opinion, it is the reason why the progress in this field is so slow. It is very clear that researches that trained only in linguistic alone cannot leverage advances in other fields that are related to natural language processing, such as Neural Machine Translation which is a part of Machine Learning, Neuroimaging which is a part of Neuroscience, Neuromorphic Chips which are part of Electronics, and Dynamical Systems which are part of Physics. The mere luck of mathematical modeling is a very constraining factor, and it propelled all fields mentioned above. That is why groups that consists of generalists that have good grasp of math, machine learning, neuroscience and engineering will be most efficient in advancing the research and practical implementation of language acquisition. 

Clearly defined goal

As Jeff Hawkins from Numenta that is focused on developing general neocortex algorithm based on neurological evidence mentioned we have enough data to work on general theory of neocortex functioning. There is no lack of data, in opposite the data is in abundance. What lacks is the clear goal of what we want to achieve and clear plan how to move into right direction. It seems to me the best approach should be something along the lines of Lunar Program back in 60th and 70th of 20th century. Though there is no need to invest billions of dollars to make a progress, but dedicated people with right background and well defined goals.

References

 

 

 

Second Language Acquisition. What do we know? Part one.

Abstract

I hope that this post will be the first one in a series of posts I want to write on the topic of second language acquisition abbreviated in linguistics as SLA. What is meant by SLA is  a language that a person learns as a second language (L2) after he had acquired the first one which is a native language (L1). The research into the subject shows that the first and second language acquisitions are interconnected and may effect each other. So it makes sense to discuss first language acquisition too.

Why am I interested in this topic?

Since childhood I was interested in how we learn languages. As my life progressed from childhood to where I am now I happened to acquire two languages with a very high level of proficiency and learned a number of others to some extent. As a native Russian speaker growing in Ukraine I learned Ukrainian as a second language at school, but my knowledge of the language is quite superficial, though I can understand it when I hear it well. Then I learned and talked Hebrew for about 19 years. Even though I also studied English back in Ukraine I never knew it well before I started to learn it by mostly reading magazines back in 1999. So I would say that real experience with English language I started to gather for about 19 years too. Though, one important point to make is that I only started to use it for speaking communication purposes for about 2 years now. In addition, back in Tel Aviv University I studied a Japanese language for a year. But my diminishing knowledge of it is rudimentary.

To summarize the above I would rate my knowledge of the languages as below, when by knowledge I understand speaking, reading and writing.

  1. Russian 
  2. Hebrew
  3. English
  4. Ukrainian
  5. Japanese

I hope that this background description explains a little bit why I might be interested in understanding how we learn a new language be it second, third or N-language.

Introduction

It is very strange that we know so little about how we learn first or second languages taking into consideration the advances in Neuroscience since early 2000 and Artificial Neural Networks starting from 2012 (also known now as Deep Learning). First, I heard about the subject of SLA back in 2004 when I studied Generative Linguistics in Tel Aviv University. When looking into the state of the art of the research back then I heard only about Noam Chomsky and Stephen Krashen’s research into this subject. Now almost 15 years since the state of the art of the research seems like frozen in the same place. But my intuition indicates, that by incorporating approaches from Supervised Machine Learning which includes Recurrent Neural networks such as LSTM and Convolutional Neural Networks with Attention Mechanism, along with a very promising research done at Numenta company and other approaches it is possible to make a significant progress in the field of second and first language acquisition.

The more detailed description of what I propose will be explained in further parts.

Welcome To The Quantum Machine

The Dawn Of Quantum Computing

There are great news for those of you who’ve heard about quantum computers and quantum computations. Finally, thanks to IBM Quantum Experience site it is possible to run experiments on a real 5 qubits Quantum Processor connecting to it via IBM Cloud interface! Isn’t this sound great?  It is an advancement in computing that is equal in its importance to the transition from mechanical to digital computers and it can be hardly overestimated. I recommend to read this IBM’s article that explains the endeavor before proceeding any further.

What Is It All About?

IBM Quantum Experience is aimed to bring hands-on skills in programming quantum processor. This is not a simple task taking into consideration that Quantum Computations far complex to grasp in comparison to classical computations and requires understanding of quantum physics and linear algebra. But it can be done with effort and dedication. In addition, there is a need in motivation and curiosity.

Prerequisites

If you have no prior knowledge of quantum computations you’d better learn a little bit at this link at IBM Quantum Computing site. In addition to this, to jump deep into experimenting with quantum algorithm there is a need to revive or acquire skills in Linear Algebra (this one is at Khan Academy) and quantum physics fundamentals concepts. Preferably, it is better to have a BSc degree in physics, engineering, mathematics etc.

How To Apply?

The application is straightforward. There is a need to register at the site. Then confirm the registration. Log in and compose your first quantum algorithm. One who has no previous knowledge in quantum computation is advised to go through very detailed guide at the site.

What Can Be Implemented On This Processor?

Well, this is exactly the question that I am asking myself right now too. I’ve found that it is possible to implement Deutsch–Jozsa algorithm and to see how it works for real. Following articles may help to understand how to do it.

What Else May Help?

There are additional resources that may come in handy tackling this topic.

Parting words

And so it begins…

 

MagLev Is Like Magic!

It’s a kind of magic

If you haven’t heard yet about Magnetic Levitation then check out the video of my final project for a BSc. in Electronics back in 2011 that did just that.

There are a number of types of magnetic levitation and you may find a lot of peculiar videos about it on YouTube. In addition, don’t miss this MagLev World Record video.

How does it work?

Specifically in my own project the maglev was implemented using digital controller that controlled the magnitude of current flowing in the electromagnet as a function of distance of a floating object from electromagnet steel core. A Hall effect sensor, actually, two of them were placed at opposite sides of electromagnet core and provided the readings of the magnetic field surrounding the electromagnet.

With a help from my friends

I’d like to thank once again Mr. Arie Shnaiderman an outstanding control engineer at ReWork company who helped me make this instrument a real thing.

 

 

 

Gravitational Waves Discovered

And so it begins …

This post is a  very special one due to the fact that on 2/11/2016 it was announced that long ago predicted from General Relativity Theory gravitational waves were detected by

Laser Interferometer Gravitational-Wave Observatory  (LIGO) in USA.

This is the most exciting discovery of the 21st century so far and its consequences are hard to predict. What is for sure that now we have a mean to probe what happens in the places such as where black holes collide.

I congratulate all of humankind with this achievement that is a great example of science teams collaboration around the world that is a way to go.

Resources you won’t want miss

  1. An interview with the Kip Thorne the genius behind the machines that made it a reality.
  2. Video from Washington’s National Science Foundation press-conference were this announcement was made.