## Profile

- Amazon.comSenior Manager - Supply Chain Control & Analytics, present
- Dixons Carphone GroupCommercial Mathematician
- Thalassa Autonomous RoboticsConsulting Systems Engineer (Navigation Algorithms R&D)
- BioSonics Inc. (Smart Sonar)Director of Engineering & Operations
- BioSonics Inc. (Smart Sonar)Head of Software Development
- Boeing Phantom Works (R&D Centre of Excellence)Scientific Programmer, NSF VIGRE Fellowship

- Google+ (Buzz) Posts (current)
- Mathematical Science & Technologies (current)
- Stack Overflow (current)
- Maths Exchange (current)
- AssadOnTech (Twitter) (current)

Over the years and on both sides of the Atlantic, I've led a variety of quantitative & operational challenges across intelligent systems & software engineering, sonar, defense, environmental technology, robotics, and most recently the complex world of multi-channel retail.

The common thread linking passion, profession, and play is applying technology, good design, and quantitative modelling and simulation, to build better products, enable better decisions, and optimise performance.

Presently, I am Commercial Mathematician (aka data scientist) to a leading UK retailer, where I develop mathematical models and algorithms that get beneath a complex of systems, processes, and behaviours, to drive better performance for customers at a lower cost. My work is applied to demand forecasting, product replenishment, supply chain optimisation, statistical modelling of multi-channel behaviour, predictive analytics, as well as decisions around property portfolio transformation, merchandise ranging & assortments, and own-brand profitability.

Prior to this, I developed navigation & localisation algorithms for unmanned autonomous underwater robotic vehicles working for the ocean robotics R&D company Thalassa Autonomous Robotics Ltd. (Bristol, UK). This neat little animation[1] illustrates the concept of what we designed for Undersea Oil & Gas Exploration. Videos of early prototypes in action are here[2], and here[3]

In the U.S., I was the Director of Engineering & Operations at BioSonics, Inc. (Seattle, Washington), where I led the design, development and manufacturing of intelligent sonar systems and their real-time software. Applications spanned a number of industries including environmental monitoring of the world's first underwater tidal energy grid in New York City's East River, homeland defense with Sandia National Labs & the Naval Underwater Warfare Centre, and off-shore aquaculture (automated fish farming) with the Chilean government, among others.

I'm always experimenting and tinkering, so feel free to get in touch with ideas.

- UW, Swarthmore

## Stream

### Assad Ebrahim

Shared publicly -**Good films about maths are rare**... but George

**Csicsery**(pronounced "Chi-cherry") has produced

**13 elegant films about mathematicians & mathematics. (List below.)**In his own words "Working with mathematicians is a great source of pleasure. It is the only group I know, where the answer 'I don't know' is met with excitement and motivation rather than with irritation. I hope that some of this excitement and passion filters through the film to audiences." [a]

Just watching the trailers is rewarding. If you've never met a mathematician, these should give you a personal look into the very diverse worlds of some great modern mathematicians and to see the humanity behind the thinking. If you're a mathematician or love mathematics, these are inspiring.

**And now, the list of 13 films, courtesy of Csicery's studio Zala Films:**

[1]

**Counting from Infinity: Yitang Zhang & the Twin Prime Conjecture**(2015)

*The central challenge of the film was finding a way to depict Yitang Zhang's dedication to working in isolation. The qualities he embraces-solitude, quiet, concentration-are the opposites of those valued in the media. Fortunately, it is a conundrum Csicsery had faced before in other films about mathematicians. He had learned that contrary to the rules, it is okay to shoot long scenes of "the grass growing," or in this case, shots of "a person just sitting with pencil and paper and thinking. The longer the scene, the more you realize that you really can see someone thinking. The human face is very expressive. Give it time and it speaks volumes."*- George Csicsery

**About**: http://www.zalafilms.com/films/countingabout.html

**Trailer**: http://www.zalafilms.com/films/countingtrailer.html

[2]

**Porridge Pulleys & Pi: Two Mathematical Journeys**(Hendrik Lenstra, Number Theory, Elliptic Curves & Cryptography) & (Vaughan Jones: Quantum Mechanics, Knot Theory, and DNA Protein Folding) (2004)

**Trailer**: http://www.imdb.com/video/wab/vi1773798425/

**About**: http://www.zalafilms.com/films/pppdirector.html

*"There are several stereotypes and beliefs about mathematicians that Porridge pulleys and Pi aims to dispel. First, I wanted to show that there is no single type of person who can become a mathematician. ... given the right training, any child with the aptitude can turn into a mathematician. ... Jones and Lenstra are the opposites of the eccentric nerdy type who has come to characterize the popular conception of what mathematicians are like. Another cliche I hope to debunk is that of the tortured genius. This film contains clear evidence that mathematicians derive a great deal of pleasure from their work."*- George Csiscery

http://www.zalafilms.com/films/pppdirector.html

[3]

**Taking the Long View: The Life of Shiing-shen Chern**(one of the fathers of modern differential geometry) (2011)

**View Short**: http://zalafilms.com/takingthelongviewfilm/viewfilm.html

**Synopsis**: http://zalafilms.com/takingthelongviewfilm/synopsis.html

*There’s a quotation from Lao Tzu, an ancient Chinese philosopher, that could have been written about Chern. ‘The master does his job and then stops. He understands that the universe is forever out of control, and that trying to dominate events goes against the current of the Tao. Because he believes in himself, he doesn’t try to convince others. Because he is content with himself, he doesn’t need other’s approval. Because he accepts himself, the whole world accepts him.’*- Alan Weinstein, UC Berkeley

*The true importance of Shiing-shen Chern’s role in the development of mathematics ... His influence with Chinese government leaders helped bring Western mathematicians to China and send Chinese students to study abroad. Today’s leaders in Chinese mathematics were all beneficiaries of Chern’s vision. His greatest contribution to the restoration of Chinese mathematics, however, is the establishment of the Nankai Institute of Mathematics, today known as the Chern Institute of Mathematics. The Chern Institute provided a base for these international interactions which often led to collaborations, reciprocal visits, and joint papers.*

About: http://www.takingthelongviewfilm.com/

*He said, ‘my policy to operate this institute is very simple. Three words in Chinese. First, no meetings. Second, no plan. Third, do more.’ That means, just do your research work.*

Molin Ge, Theoretical Physicist, Chern Institute of Mathematics

[4]

**Invitation to Discover**: An Introduction to the MSRI (Mathematical Sciences Research Institute) (2002)

**Watch**: http://www.msri.org/web/msri/online-videos/special-productions-events/invitation-to-discover

[5]

**I Want to Be a Mathematician: A Conversation with Paul Halmos**(2009)

**Trailer**: http://www.zalafilms.com/films/halmostrailer.mov

**About**: http://www.zalafilms.com/films/halmossynopsis.html

*"When an engineer knocks at your door with a mathematical question, you should not try to get rid of him or her as quickly as possible. You are likely to make a mistake I myself made for many years: to believe that the engineer wants you to solve his or her problem. This is a kind of over simplification for which mathematicians are notorious. Believe me, the engineer does not want you to solve his or her problem. Once I did so by mistake (actually I had read the solution in the library two hours previously, quite by accident) and he got quite furious, as if I were taking away his livelihood. What an engineer wants is to be treated with respect and consideration, like the human being he is, and most of all to be listened to with rapt attention. If you do this, he will be likely to hit upon a clever idea as he explains the problem to you, and you will get some of the credit. Listening to engineers and other scientists is our duty. You may learn some interesting mathematics while doing so."*- Gian-Carlo Rota, Indiscreet Thoughts 1979

[6]

**Julia Robinson and Hilbert's 10th Problem**(2008)

**Trailer**: http://www.zalafilms.com/films/julia.html

**About**: http://www.zalafilms.com/films/jrbackground.html

[7]

**Navajo Math Circles**(in production)

*"Open-ended questions are totally new to most of the kids. Usually you have to have an answer within 20 seconds, 30 seconds, that’s what math is. Math circles are the opposite. We start with some simple questions, and we ask more questions and more questions. We get some answers along the way. The answers actually don’t matter. The more and more questions… we’re opening whole research problems, and that is something totally new to the kids. And once they like it, it’s just amazing, it’s transformative.*" - Matthias Kawski, Arizona State University

**Trailer**: http://www.zalafilms.com/navajo/trailer.html

**About**: http://www.zalafilms.com/navajo/about.html

[8]

**Hard Problems: The Road to World's Toughest Math Contest**, covering the story of the 2006 US IMO team (2008)

*Trailer: http://www.hardproblemsmovie.com/trailer.html

**About**: http://www.hardproblemsmovie.com/synopsis.html

[9]

**N is a Number: A Portrait of Paul Erdos**(1993)

**Trailer**: https://www.simonsfoundation.org/multimedia/n-is-a-number-a-portrait-of-paul-erdos/

**About**: http://www.zalafilms.com/films/nisfilm.html

**Full**: https://www.youtube.com/watch?v=wN4yLPPvRBgs

[10]

**Erdos 100**(2013)

**Trailer**: http://www.zalafilms.com/films/erdostrailer.html

**About**: http://www.zalafilms.com/films/erdos.html

[11]

**To Prove and Conjecture: Excerpts from Three Lectures**by Paul Erdos (1993)

**About**: http://www.zalafilms.com/films/prove.html

[12]

**The Right Spin: How to fly a broken space craft**(Mir), the Story of a Dramatic Rescue in Space and the Mathematics Behind It (2005)

**About**: http://plus.maths.org/content/right-spin-how-fly-broken-space-craft http://archive.msri.org/specials/rightspin

**Alternate documentary**: https://www.youtube.com/watch?v=tM7fTLLmgbk

[13]

**On Mathematical Grounds**: A Refresh of an Introduction to the Mathematical Sciences Research Institute (MSRI) (2009)

**About**: http://www.zalafilms.com/films/omg.html

**YouTube**: https://www.youtube.com/watch?v=MO-JRzTiSNo

**References**:

[a]

**George Csicsery**, Producer & Director, Zala Films, in his own words: http://zalafilms.com/takingthelongviewfilm/directors_statement.html

[b]

**The Films**: http://www.zalafilms.com/films/index.html

[c]

**The story of George Csicsery**: Math Films, Yes - But So Much More: http://cinesourcemagazine.com/index.php?/site/comments/csicsery_math_films_yes_but_so_much_more/

[d]

**Science Lives**, Simon Foundation & Zala Films, videos of interviews with living mathematicians https://www.simonsfoundation.org/category/multimedia/science-lives/alphabetical-listing/

### Assad Ebrahim

• Mathematics -**The Three Temperaments of Mathematicians**This piece explores the three intellectual temperaments, characterised as

**birds, frogs, and beavers**, from the point of view of famous mathematicians.

**Article**: http://lesswrong.com/lw/2z7/draft_three_intellectual_temperaments_birds_frogs/

### Assad Ebrahim

Shared publicly -**Time & Knowledge also need Support with the Wallet**If you've benefited from Wikipedia in your researches, please consider giving today.

Remember days gone by when one would have to pay to get to the nearest research library, then pay again to photocopy what was needed.

The digital dissemination of knowledge, though freely given, does require our support.

Please do your part now in this annual campaign. The average donation is $10.

### Assad Ebrahim

• General/Interdisciplinary -**The Evolution of Standards**The modern notion of measurements based upon invariable units of nature was largely unknown until the dawn of the 1800s. Prior to this, measurement standards were mostly non-reproducible, tracing their origins by convention back to medieval measures "based on the size of barley corns and the length of human feet" [1; 527]

The image below shows a typical example: the

**surveying rod**was conventionally defined to be "sixteen feet".

**But whose sixteen feet?**One standard specification from 1536 was clear:

*"Take sixteen men, short men and tall ones as they leave church and let each of them put one shoe after the other and the length thus obtained shall be a just and common measuring rod to survey the land with."*(1536; Geometry; Jacob Kobel) [1; 48]

Reference:

[1] Measures for Progress: A History of the National Bureau of Standards; Rexmond C. Cochrane; 1966, 1974, National Bureau of Standards, US Dept of Commerce;

Available for download from:

http://www.nist.gov/nvl/upload/Measures_for_Progress-MP275-FULL.pdf

#historyofscience #measurement #standardization

2

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**What standard unit of measure would permit working readily with this wide dynamic range of measurements?**

The decibel, which is the practical man's logarithms, and appears everywhere in oceanography.

There's a great quote by Laplace, whose tribute to the logarithm was much more glowing than the typical student reaction:

"[Logarithms: that] admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations."

Substitute oceanographer for astronomer and the sentiment, I think, remains the same.

### Assad Ebrahim

Shared publicly -**The mechanics of mosquito-born illness.**Wondering whether that mosquito bite could hold an unpleasant E*-surprise? The key is that parasites have to have evolved

**specific mechanisms**to allow the mosquito to be used as a carrier. So in a sense it is evolution itself that protects us from the mosquito becoming a universal plague-bearing vector. This fascinating half-pager also explains why the Ebola virus is highly unlikely to be transmitted by mosquitoes in the near future. Getting bitten by mosquitos? Don't worry, you're not going to get Ebola that way. #ebola #evolution

### Assad Ebrahim

Shared publicly -**Ebola: how culture & politics have combined to allow mathematics to drive the epidemic.**Three parameters, amongst several, influence the dynamics of contagion:

**(1) incubation period**, which determines how far an infected individual can travel symptom-free before succumbing.

*(Ebola: 1-3 weeks)*

**(2) shedding period**, or how long an infected individual can pass on the infection to others.

*(Ebola: from first symptoms to even*

*after**death.)*

**(3) fatality rate**, or the likelihood of death once the infection has been contracted.

*(Ebola: greater than 60%, up to 90%.)*

For Ebola, these three parameters mean a plague that is capable of out-running the limited infrastructure that has currently been mobilised against it.

This sobering article reviews the rise of the current outbreak of Ebola, and how geography, culture, and politics combined to set the stage for a potential nightmare scenario.

References:

[1] How Ebola Sped out of Control, Washington Post, 4 Oct, 2014

http://www.washingtonpost.com/sf/national/2014/10/04/how-ebola-sped-out-of-control/

[2] The Ominous Math of the Ebola Epidemic, Washington Post, 9 Oct, 2014

http://www.washingtonpost.com/national/health-science/the-ominous-math-of-the-ebola-epidemic/2014/10/09/3cad9e76-4fb2-11e4-8c24-487e92bc997b_story.html

### Assad Ebrahim

Shared publicly -**Mathematical Virology**: using cryptographic techniques to unlock the code governing how viruses reproduce.

**Mathematician Reidun Twarock**(U. York) describes: "The Enigma machine metaphor is apt.

**The first observations pointed to the existence of some sort of a coding system**, so we set about deciphering the cryptic patterns underpinning it using novel, purpose designed computational approaches. We found multiple dispersed patterns working together in an incredibly intricate mechanism and we were eventually able to unpick those messages. We have now proved that those computer models work in real viral messages." [1]

**Paper**: Revealing the density of encoded functions in a viral RNA, Proceedings of the National Academy of Sciences, 2015

http://www-users.york.ac.uk/~rt50/JournalPublications_2014.html

**Reidun Twarock**: http://maths.york.ac.uk/www/rt507

**A Course in Mathematical Virology**: http://maths.york.ac.uk/www/node/14102

**Source**:

[1] http://www.sciencedaily.com/releases/2015/02/150204075224.htm

### Assad Ebrahim

Shared publicly -**Recommender systems**are a hot topic in mathematical applications to

**retail and search**. There are various approaches, using mathematical techniques ranging from Matrix factorisation to PCA methods for similarity determination, from Classifiers and Collaborative filters, to Decision trees and K-Nearest-Neighbours.

As should be expected given the wide variation in application areas, there is no "best" approach that works across all possible problem domains -- what works pragmatically is to get started, tune your recommender agent to your particular problem domain, and then see how well it performs against either practical performance measures (e.g. improved conversion) or theoretical measures of error on benchmark training data sets (e.g. the Netflix prize)

Some reasonable mathematical understanding is needed to feel comfortable with the algorithms, as most use applied mathematics, modern statistics or machine learning techniques.

However, neither a strong mathematical background nor previous software development expertise is needed to simply try out pre-existing algorithms using "canned packages" and to play around with various data sets. This is a good way to build up some intuition if you have never seen a recommender algorithm in action. This intuition will then help as you begin the process of building up the knowledge necessary to understand the technical details.

I am often asked how to get started in this area of applied mathematics. Below are some general responses with references that should get you started should you be interested.

**Question 1: Where can I get rich data sets from real world situations?**

There are free data sets available from a variety of sources. One good place to start is

**RecSysWiki**[1]

Another is the competition site Kaggle. [2]

**Question 2: What programming language should I use to give these ideas a try?**

Use any language you know well! Python is a useful modern software language that is well worth knowing as it has a large collection of well designed numerical libraries. R is another popular choice, especially if you're interested in statistical applications and machine learning more generally.

Indeed, many languages will have existing canned packages offering recommender systems.

R has recommenderlab [3], and yhat has written a nice blog article on the subject. [8]

Python has

**python-recsys**[4]

Ruby and Perl also have entries.

So, if you want something pre-canned, you can probably find something in just about any reasonably modern language. The more important question will be: can you find something that works for

**you**in that language. And if not, should you "roll your own"...

**Question 3: Should I use a pre-canned algorithm or "roll my own"?**

For your

**learning**about algorithms and implementations, roll your own. There's no better way to understand an algorithm than to attempt to write your own implementation.

For a production grade implementation however, it's probably wiser to work with a reputable package that has a wide installed user base, so that you can be sure that most of the bugs have been worked out, that the maths is sound, and that any updates will be made available to you.

**Question 4: Which algorithms should I focus on? What about X?**

There are at least a dozen popular algorithms that you could consider. I will not cover all of them here.

But I often get asked about Neural Networks in recommender systems. My personal opinion is that neural networks are more work than they are worth. They are often tuned very specifically to specific training data, and don't have a general theory behind why they are or are not effective, beyond the fact that neural networks are a form of multi-dimensional piecewise linear approxiation, which means a neural network can in principle be made to work well with anything, by introducing enough neurons.

My recommendation is to steer clear from Neural Networks when you're first starting out. Matrix factorisations or other direct measures of similarity are much more appealing (in my opinion) both on theoretical grounds and on their practical performance.

**Question 5: Are there some useful papers I can start with to get a good overview of the subject?**

Yes. Here's three that I think would have something useful for someone entering the subject at a variety of levels. All three great papers are freely available from the internet. A Google search on "Recommender Systems" will turn up a number of others.

* Matrix Factorisation Techniques for Recommender Systems (2009) [5]

* Mining of Massive Data Sets, Ch.9: Recommender Systems (2010) [6]

* Tutorial: Recommender Systems (Slides) (2013) [7]

Good luck! and feel free to write about your learnings in the comments.

**References:**

[1] Datasets from RecSysWiki:

http://www.recsyswiki.com/wiki/Category:Dataset

[2] Kaggle

http://www.kaggle.com

[3] RecommenderLab (R)

http://cran.r-project.org/web/packages/recommenderlab/

[4] Python-Recsys:

http://ocelma.net/software/python-recsys/build/html/quickstart.html

[5] Matrix Factorisation Techniques for Recommender Systems; Yehuda Koren, Robert Bell, and Chris Volinksy; August 2009; IEEE;

http://www2.research.att.com/~volinsky/papers/ieeecomputer.pdf

[6] Mining of Massive Data Sets, Ch.9: Recommender Systems; Jure Leskovec; Anand Rajamaran; Jeffrey Ullman; 2010; URL: http://www.mmds.org/ and http://infolab.stanford.edu/~ullman/mmds/ch9.pdf

[7] Tutorial: Recommender Systems (Slides); August 2013; International Joint Conference on Artificial Intelligence; Beijing; Dietmar Jannach and Gerhard Friedrich

Source: ijcai13.org/files/tutorial_slides/td3.pdf

[8] Tutorial: Building a Recommendation System in R; July 2013; yhat

http://blog.yhathq.com/posts/recommender-system-in-r.html

*Image Credit: cover of the book Recommender Systems: An Introduction, by Dietmar Jannach*

*et.al*

*.*

2

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### Assad Ebrahim

Shared publicly -**MetaMoji Note: A Digital Pencil & Notebook Replacement**Notebook & Pencil are hard to beat when sketching, equations, and writing are required in equal measure. To be sure, there are good digital tools for each of these separately, but not when all three are combined, which is the typical use case when working as an engineer, mathematician, or product designer.

Now there's a new suite of digital freehand tools that are almost good enough to set aside the notebook permanently. The core of this toolkit is a vector-based graphics canvas (software) and an ultra-precise stylus (hardware).

If you use Apple iOS, you may already be using Adobe Ideas, which does the job brilliantly and is free. [1] But Adobe has no plans to release this to either Windows RT or Android, so for the rest of us, that's a showstopper.

Enter

**MetaMoji Note**, a comparable vector-graphic app that works seamlessly across all three major platforms (

**iOS, Windows RT, and Android**) with its free cloud sync feature. There are several attractions. First,

**Note**has that simple, intuitive, easy-to-use interface that starts to rival the simplicity of pen & paper. Additionally, it has virtual whiteboard technology for collaborative work (Share Anytime), and exceptional hand-writing recognition with the mazec3 add-on.

Initial drafts are a snap. Refined editing / re-work is easy. And sprucing up for discovery sessions / presentations is not much more work. I now use

**Note**as my primary digital notepad.

For usability, I combine finger motions on my touchscreen laptop with a precise stylus from Wacom which can also be fitted with a wireless module to go cable free.

The results have been good enough to set aside my notebook & pencil for weeks at a time.

An excellent starting point to freehand diagramming, with tips and examples, is +Simon Raper's article [1] available from Drunks & Lampposts.

Kit List:

+ MetaMoji Note (£5, outstanding) Note Lite (Free, still very good)

+ Wacom Intuos Pen (small)

+ Wacom Wireless Accessory (optional)

+ Touchscreen Laptop with adequate processing power (Intel Core i5 with 6GB RAM is plenty)

+ Windows 8.1

References:

[1] Freehand Diagrams with Adobe Ideas, Jan 2014, Simon Raper, Drunks & Lampposts (blog)

http://drunks-and-lampposts.com/2014/01/13/freehand-diagrams-with-adobe-ideas/

[2] MetaMoji Note (iOS, Windows RT, & Android) (£5)

http://noteanytime.com/en/

[3] Wacom Intuos Pen (CTL-480S) (£49)

http://www.wacom.com/en/de/creative/intuos-pen

[4] Adobe Ideas (iOS only) (Free)

https://itunes.apple.com/us/app/adobe-ideas/id364617858?mt=8&affId=1503186

[5] Lenovo Flex 2 laptop, with touchscreen display (1920x1080), Intel Core i5-4210P processor with 6GB RAM, 1.7 GHz and 3MB Cache (£429)

http://shop.lenovo.com/gb/en/laptops/lenovo/flex/flex-2-14/#tab-tech_specs

#productivity #innovation #technology

Here's a quick list & a few tips:

* Cloud synch (free storage). This means data safety and accessibility from any device.

* Pen selection: each pen style has particular characteristics.

+ The fountain pen assists with neater handwriting for those with a scrawl.

+ The brush improves annotations.

+ Colours enliven sketches & diagrams.

* Paper selection: lined, graph, boxes, grids, and a variety of others

* Pen Tip size affects your handwriting at different zoom settings.

+ I typically use 5pt for primary writing, one line per ruled line (this keeps the right size for legible print-outs); 3pt for fine annotations / commentary; 10pt for brush annotations (large, e.g. numbering)

* Lasso for partial selection: There are two choices of lasso's - complete selection and partial selection. With a bit of practice, you'll find partial selection is exactly what you need to grab whatever bits you want.

* Rotation, reflection, resize with dynamic thickness, resize without changing thickness

+ Text should be resized dynamically, so when you shrink, the stroke thins and retains legibility, and when you enlarge again, the stroke thickens restoring printability & aesthetics.

* Handwriting recognition (mazec): this is an add-on. Its recognition algorithms are exceptional. So on smaller devices, mazec offers a pretty good alternate input device, comparable to Swype on-screen keyboards.

However, as an add-on for your laptop installation, my feelings are mixed:

Probably worth having if you write faster than you type.

In my case, I can type much faster than I write, and indeed, when I choose to write it is because I

**want**the deliberateness & time that writing gives one to reflect. Furthermore, I want to preserve the visual arrangement that writing / sketching allow uniquely.

As you can see, MetaMoJi Note has much to offer.

Happy exploring!

### Assad Ebrahim

Shared publicly -**How to Write a 21st Century Proof?**If you've gone through proofs in published papers, you'll know it can be hard work to convince yourself that what's being claimed is in fact true. And it's not uncommon to conclude that whilst the claim is indeed correct, the presented proof is not. [3,1]

Mathematician turned Computer Scientist, Leslie Lamport, suggests an

**hierarchical structured method**for writing proofs that both improves their rigour

**and**eases their communication.

At first glance, the Lamport structured method may seem like a radical departure from the usual prose paragraph method. But after giving it a try, I've found it to be refreshingly freeing and have come to prefer it for my own work.

The common prose-style methods in contrast seem to be more appropriately labelled "proof sketch".

Now there's certainly a place for proof sketches. Indeed every non-trivial proof benefits from a prose overview. But the problem with proof sketches being the method of publishing proof is that "authors seem to choose randomly which details to supply and which to omit." [1:p10]

How many errors are there in published mathematical proofs?

"A previous editor of the Mathematical Reviews [remarked] that approximately one half of the proofs published in it were incomplete and/or contained errors, although the theorems they were purported to prove were essentially true." [3, p.71]

That's a claim which suggests Lamport's methods may indeed have something valuable to offer.

**Try the Structured Method**

To give Lamport's structured method a try, read one of his short papers available from the links given below.

Then I suggest a "test drive" with the following three problems. All three are elementary in the sense that the content of each should hold no trouble for most of you. The issue will be what

*you*think is a sufficiently convincing proof now that you've read his critique.

**Test Problems**

*Declare whether the statement is True or False. If False, produce a counterexample. If True, write down a complete proof.*

(1) The intersection of any two intervals is an interval.

(2) There is always a real solution to the equation y^2 - a = 0, when a>0. (For simplicity, think of a=2.)

(3) N(X+Y) = N(X) + N(Y) - N(X.Y), where X and Y are each finite sets, + is set union, . is set intersection, and N( ) is the number of elements contained in the set.

**Lamport's Papers**

[1995]

**How to Write a Proof**; Leslie Lamport; 1995; Aug-Sep, American Mathematical Monthly, Vol 102 No 7 pp.600-608

**PDF here**: http://research.microsoft.com/en-us/um/people/lamport/pubs/pubs.html#lamport-how-to-write

[2012]

**How to Write a 21st Century Proof**; Leslie Lamport; 2012; March; Journal of Fixed Point Theory and Applications

**PDF here**: http://research.microsoft.com/en-us/um/people/lamport/pubs/pubs.html#proof

**Other References**

[1983] Rigorous Proof in Mathematics Education; Gila Hanna; 1983; OISE Press (University of Toronto)

### Assad Ebrahim

Shared publicly -This summer, +Nick Hanauer wrote in Politico Magazine : "The Pitchforks are Coming: For Plutocrats". His message was blunt. Unless far reaching structural changes are accepted and pushed through

*by the wealthy*to redress the rampant inequality of our economic structures, then change will likely come suddenly --- as it often has in history --- on the end of a pitchfork. [1]

The warning is not new. For the third year running, the annual World Economic Forum's survey of 700 global experts placed income disparity at the top of the list of key global risks. [2]

Now, the same warning is being sounded again, from the U.S. Federal Reserve. This carefully worded address by Chairman Janet Yellen [3] has some startling facts:

* The average net worth of an American today is $11,000, exactly

*half*the average net worth 25 years ago, in 1989, of $22,000.

* Amongst those with children, the average net worth today is even lower, at $8,000.

Turning to wealth distribution:

* The bottom 50% of Americans together hold just 1% of the nation's total wealth. What this means is that 99% of the nation's total wealth is held by the top 50%.

* The reality, however, is even worse: the top 10% of the top half hold a staggering 63% of the nation's total wealth.

Personally, I cannot see how current trends are sustainable. But I also cannot see how they can be reversed without a fundamental change in our economic structures, and what it means to work, earn, live, share, and prosper -- together.

History has shown time and again that nothing, in the grand scheme of things is "too big to fail". The problem is that what seems to fail, and fail again, is our determination to

**ensure**that we raise up the prosperity of others even as we ourselves are raised.

Perhaps Marx had some of it right: some of the rising is on the backs of others, which introduces into the argument an element of exploitation.

But for completeness, let's at least reference the other side of the argument: Paul Graham writes an articulate opposing view to the above sentiment, explaining why income inequality per se is not the problem that should be tackled.

See what you think.

**References:**

[1] The Pitchforks are Coming, Nick Hanaer, Politico Magazine, June 2014, https://plus.google.com/u/0/+AssadEbrahim/posts/RBVukVgfe3e

[2] Income Inequality tops global risks, LA Times, Jan 2014,

http://articles.latimes.com/2014/jan/17/business/la-fi-mo-world-economic-forum-income-inequality-20140117

[3] Rising Inequality, New Yorker Magazine, Oct 2014

http://www.newyorker.com/news/john-cassidy/janet-yellen-tells

[4] Inequality and Risk, Paul Graham, Aug 2005

http://www.paulgraham.com/inequality.html

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### Assad Ebrahim

Shared publicly -**The Eight Great UK Technologies**The UK last year set out a clear and ambitious 10 year technology plan (to 2023) for developing World Leading Technology Centres. There are three highlights to look for in this wide-ranging address. The first is the contrast between the UK and the US, the second is the discussion of the Eight Great Technologies that will be the primary focus of the UK's technology advancement programmes, and, finally, and perhaps most interestingly for a political address, is the acknowledgement and discussion of the six possible outcomes of this 10 year plan, everything from fizzle (false dawn) to fantastic (home to world class R&D centres).

The Eight Great Technologies (the UK's Focus)

1. Big Data

2. Space

3. Robotics and Autonomous Systems

4. Synthetic Biology

5. Regenerative Medicine

6. Agri-Science

7. Advanced Materials

8. Energy

The Six Possible Outcomes:

1. False Dawn

2. Transmutation

3. Gone abroad

4. It's here but it isn't ours

5. We have grown big new companies

6. We are purveyors of R&D to the world

Source:

Address made on 24 Jan 2013 by the Rt Hon David Willetts MP, Dept for Business, Innovation & Skills

https://www.gov.uk/government/speeches/eight-great-technologies

#innovation #uktech