December 06, 2016

November 17, 2016

William Stein

RethinkDB, SageMath, Andreessen-Horowitz, Basecamp and Open Source Software

RethinkDB and sustainable business models

Three weeks ago, I spent the evening of Sept 12, 2016 with Daniel Mewes, who is the lead engineer of RethinkDB (an open source database). I was also supposed to meet with the co-founders, Slava and Michael, but they were too busy fundraising and couldn't join us. I pestered Daniel the whole evening about what RethinkDB's business model actually was. Yesterday, on October 6, 2016, RethinkDB shut down.

I met with some RethinkDB devs because an investor who runs a fund at the VC firm Andreessen-Horowitz (A16Z) had kindly invited me there to explain my commercialization plans for SageMath, Inc., and RethinkDB is one of the companies that A16Z has invested in. At first, I wasn't going to take the meeting with A16Z, since I have never met with Venture Capitalists before, and do not intend to raise VC. However, some of my advisors convinced me that VC's can be very helpful even if you never intend to take their investment, so I accepted the meeting.

In the first draft of my slides for my presentation to A16Z, I had a slide with the question: "Why do you fund open source companies like RethinkDB and CoreOS, which have no clear (to me) business model? Is it out of some sense of charity to support the open source software ecosystem?" After talking with people at Google and the RethinkDB devs, I removed that slide, since charity is clearly not the answer (I don't know if there is a better answer than "by accident").

I have used RethinkDB intensely for nearly two years, and I might be their biggest user in some sense. My product SageMathCloud, which provides web-based course management, Python, R, Latex, etc., uses RethinkDB for everything. For example, every single time you enter some text in a realtime synchronized document, a RethinkDB table gets an entry inserted in it. I have RethinkDB tables with nearly 100 million records. I gave a talk at a RethinkDB meetup, filed numerous bug reports, and have been described by them as "their most unlucky user". In short, in 2015 I bet big on RethinkDB, just like I bet big on Python back in 2004 when starting SageMath. And when visiting the RethinkDB devs in San Francisco (this year and also last year), I have said to them many times "I have a very strong vested interest in you guys not failing." My company SageMath, Inc. also pays RethinkDB for a support contract.

Sustainable business models were very much on my mind, because of my upcoming meeting at A16Z and the upcoming board meeting for my company.  SageMath, Inc.'s business model involves making money from subscriptions to SageMathCloud (which is hosted on Google Cloud Platform); of course, there are tons of details about exactly how our business works, which we've been refining based on customer feedback. Though absolutely all of our software is open source, what we sell is convenience, easy of access and use, and we provide value by hosting hundreds of courses on shared infrastructure, so it is much cheaper and easier for universities to pay us rather than hosting our software themselves (which is also fairly easy). So that's our business model, and I would argue that it is working; at least our MRR is steadily increasing and is more than twice our hosting costs (we are not cash flow positive yet due to developer costs).

So far as I can determine, the business model of RethinkDB was to make money in the following ways: 1. Sell support contracts to companies (I bought one). 2. Sell a closed-source proprietary version of RethinkDB with extra features that were of interest to enterprise (they had a handful of such features, e.g., audit logs for queries). 3. Horizon would become a cloud-hosted competitor to Firebase, with unique advantages that users have the option to migrate from the cloud to their own private data center, and more customizability. This strategy depends on a trend for users to migrate away from the cloud, rather than to it, which some people at RethinkDB thought was a real trend (I disagree).

I don't know of anything else they were seriously trying right now. The closed-source proprietary version of RethinkDB also seemed like a very recent last ditch effort that had only just begun; perhaps it directly contradicted a desire to be a 100% open source company?

With enough users, it's easier to make certain business models work. I suspect RethinkDB does not have a lot of real users. Number of users tends to be roughly linearly related to mailing list traffic, and the RethinkDB mailing list has an order of magnitude less traffic compared to the SageMath mailing lists, and SageMath has around 50,000 users. RethinkDB wasn't even advertised to be production ready until just over a year ago, so even they were telling people not to use it seriously until relatively recently. The adoption cycle for database technology is slow -- people wisely wait for Aphyr's tests, benchmarks comparing with similar technology, etc. I was unusual in that I chose RethinkDB much earlier than most people would, since I love the design of RethinkDB so much. It's the first database I loved, having seen a lot over many decades.

Conclusion: RethinkDB wasn't a real business, and wouldn't become one without year(s) more runway.

I'm also very worried about the future of RethinkDB as an open source project. I don't know if the developers have experience growing an open source community of volunteers; it's incredibly hard and its unclear they are even going to be involved. At a bare minimum, I think they must switch to a very liberal license (Apache instead of AGPL), and make everything (e.g., automated testing code, documentation, etc) open source. It's insanely hard getting any support for open source infrastructure work -- support mostly comes from small government grants (for research software) or contributions from employees at companies (that use the software). Relicensing in a company friendly way is thus critical.

Company Incentives

Companies can be incentived in various ways, including:
  • to get to the next round of VC funding
  • to be a sustainable profitable business by making more money from customers than they spend, or
  • to grow to have a very large number of users and somehow pivot to making money later.
When founding a company, you have a chance to choose how your company will be incentived based on how much risk you are willing to take, the resources you have, the sort of business you are building, the current state of the market, and your model of what will happen in the future.

For me, SageMath is an open source project I started in 2004, and I'm in it for the long haul. I will make the business I'm building around SageMathCloud succeed, or I will die trying -- therefore I have very, very little tolerance for risk. Failure is not an option, and I am not looking for an exit. For me, the strategy that best matches my values is to incentive my company to build a profitable business, since that is most likely to survive, and also to give us the freedom to maintain our longterm support for open source and pure mathematics software.

Thus for my company, neither optimizing for raising the next round of VC or growing at all costs makes sense. You would be surprised how many people think I'm completely wrong for concluding this.

Andreessen-Horowitz

I spent the evening with RethinkDB developers, which scared the hell out of me regarding their business prospects. They are probably the most open source friendly VC-funded company I know of, and they had given me hope that it is possible to build a successful VC-funded tech startup around open source. I prepared for my meeting at A16Z, and deleted my slide about RethinkDB.

I arrived at A16Z, and was greeted by incredibly friendly people. I was a little shocked when I saw their nuclear bomb art in the entry room, then went to a nice little office to wait. The meeting time arrived, and we went over my slides, and I explained my business model, goals, etc. They said there was no place for A16Z to invest directly in what I was planning to do, since I was very explicit that I'm not looking for an exit, and my plan about how big I wanted the company to grow in the next 5 years wasn't sufficiently ambitious. They were also worried about how small the total market cap of Mathematica and Matlab is (only a few hundred million?!). However, they generously and repeatedly offered to introduce me to more potential angel investors.

We argued about the value of outside investment to the company I am trying to build. I had hoped to get some insight or introductions related to their portfolio companies that are of interest to my company (e.g., Udacity, GitHub), but they deflected all such questions. There was also some confusion, since I showed them slides about what I'm doing, but was quite clear that I was not asking for money, which is not what they are used to. In any case, I greatly appreciated the meeting, and it really made me think. They were crystal clear that they believed I was completely wrong to not be trying to do everything possible to raise investor money.

Basecamp

During the first year of SageMath, Inc., I was planning to raise a round of VC, and was doing everything to prepare for that. I then read some of DHH's books about Basecamp, and realized many of those arguments applied to my situation, given my values, and -- after a lot of reflection -- I changed my mind. I think Basecamp itself is mostly closed source, so they may have an advantage  in building a business. SageMathCloud (and SageMath) really are 100% open source, and building a completely open source business might be harder. Our open source IP is considered worthless by investors. Witness: RethinkDB just shut down and Stripe hired just the engineers -- all the IP, customers, etc., of RethinkDB was evidently considered worthless by investors.

The day after the A16Z meeting, I met with my board, which went well (we discussed a huge range of topics over several hours). Some of the board members also tried hard to convince me that I should raise a lot more investor money.

Will Poole: you're doomed

Two weeks ago I met with Will Poole, who is a friend of a friend, and we talked about my company and plans. I described what I was doing, that everything was open source, that I was incentivizing the company around building a business rather than raising investor money. He listened and asked a lot of follow up questions, making it very clear he understands building a company very, very well.

His feedback was discouraging -- I said "So, you're saying that I'm basically doomed." He responded that I wasn't doomed, but might be able to run a small "lifestyle business" at best via my approach, but there was absolutely no way that what I was doing would have any impact or pay for my kids college tuition. If this was feedback from some random person, it might not have been so disturbing, but Will Poole joined Microsoft in 1996, where he went on to run Microsoft's multibillion dollar Windows business. Will Poole is like a retired four-star general that executed a successful campaign to conquer the world; he been around the block a few times. He tried pretty hard to convince me to make as much of SageMathCloud closed source as possible, and to try to convince my users to make content they create in SMC something that I can reuse however I want. I felt pretty shaken and convinced that I needed to close parts of SMC, e.g., the new Kubernetes-based backend that we spent all summer implementing. (Will: if you read this, though our discussion was really disturbing to me, I really appreciate it and respect you.)

My friend, who introduced me to Will Poole, introduced me to some other people and described me as that really frustrating sort of entrepreneur who doesn't want investor money. He then remarked that one of the things he learned in business school, which really surprised him, was that it is good for a company to have a lot of debt. I gave him a funny look, and he added "of course, I've never run a company".

I left that meeting with Will convinced that I would close source parts of SageMathCloud, to make things much more defensible. However, after thinking things through for several days, and talking this over with other people involved in the company, I have chosen not to close anything. This just makes our job harder. Way harder. But I'm not going to make any decisions based purely on fear. I don't care what anybody says, I do not think it is impossible to build an open source business (I think Wordpress is an example), and I do not need to raise VC.

Hacker News Discussion: https://news.ycombinator.com/item?id=12663599

Chinese version: http://www.infoq.com/cn/news/2016/10/Reflection-sustainable-profit-co

by William Stein ([email protected]) at November 17, 2016 03:57 PM

November 06, 2016

Lauren Devitt

Logicomix- logic by number

A book review of Logicomix. Written by Apostolos Doxiadis and Christos H. Papadimitriou. Art by Alecos Papadatos and Annie Di Donna.  This is the first graphic novel I've ever read. I had no idea mathe history graphic novels was a genre. I've justs tarted reading my second about Ada lovelace and Charles Babbage. Logicomix is primarily abourt Bertrand Russell, creator of russells paradox. it is written from the view of the authors and artisits of the novel as well as switching to whats going on

November 06, 2016 12:00 AM

November 05, 2016

Lauren Devitt

Improbable by Number

As a Mathematician I enjoy reading non-fiction that is not too "mathy", one that makes subjects such as probability approachable. This is the first in a series of reviews of books I've been reading for fun and for my math history course. These reviews will high light some of the more interesting parts of the book as well as rate how approachable the book made the subject and how educational on the subject the books really are. The first book I am Reviewing is The Improbability Principle  by

November 05, 2016 12:00 AM

October 11, 2016

Lauren Devitt

Knots by Number, Quipu

For my undergraduate thesis my focus was in knot theory, a subset of topology and a newer subject in mathematics. These are not the knots we think of in our daily life. The knots I studied had no ends and no thickness. Knots as we know them in everyday life have been around since before the Greeks. The book of Kells is decorated with intricate knot work. Sailors have used them on their ships. The Incas even used them to keep track of their accounting. The Incan civilization started in modern day

October 11, 2016 12:00 AM

October 10, 2016

William Stein

RethinkDB must relicense NOW

What is RethinkDB?

RethinkDB is a INCREDIBLE high quality polished open source realtime database that is easy to deploy, shard, replicate, and supports a reactive client programming model, which is useful for collaborative web-based applications. Shockingly, the 7-year old company that created RethinkDB has just shutdown. I am the CEO of a company, SageMath, Inc., that uses RethinkDB very heavily, so I have a strong interest in RethinkDB surviving as an independent open source project.

Three Types of Open Source Projects

There are many types of open source projects. RethinkDB was the type of open source project where most work on RethinkDB has been fulltime focused work, done by employees of the RethinkDB company. RethinkDB is licensed under the AGPL, but the company promised to make the software available to customers under other licenses.

Academia: I started the SageMath open source math software project in 2005, which has over 500 contributors, and a relatively healthy volunteer ecosystem, with about hundred contributors to each release, and many releases each year. These are mostly volunteer contributions by academics: usually grad students, postdocs, and math professors. They contribute because SageMath is directly relevant to their research, and they often contribute state of the art code that implements algorithms they have created or refined as part of their research. Sage is licensed under the GPL, and that license has worked extremely well for us. Academics sometimes even get significant grants from the NSF or the EU to support Sage development.

Companies: I also started the Cython compiler project in 2007, which has had dozens of contributors and is now the defacto standard for writing or wrapping fast code for use by Python. The developers of Cython mostly work at companies (e.g., Google) as a side project in their spare time. (Here's a message today about a new release from a Cython developer, who works at Google.) Cython is licensed under the Apache License.

What RethinkDB Will Become

RethinkDB will no longer be an open source project whose development is sponsored by a single company dedicated to the project. Will it be an academic project, a company-supported project, or dead?

A friend of mine at Oxford University surveyed his academic CS colleagues about RethinkDB, and they said they had zero interest in it. Indeed, from an academic research point of view, I agree that there is nothing interesting about RethinkDB. I myself am a college professor, and understand these people! Academic volunteer open source contributors are definitely not going to come to RethinkDB's rescue. The value in RethinkDB is not in the innovative new algorithms or ideas, but in the high quality carefully debugged implementations of standard algorithms (largely the work of bad ass German programmer Daniel Mewes). The RethinkDB devs had to carefully tune each parameter in those algorithms based on extensive automated testing, user feedback, the Jepsen tests, etc.

That leaves companies. Whether or not you like or agree with this, many companies will not touch AGPL licensed code:
"Google open source guru Chris DiBona says that the web giant continues to ban the lightning-rod AGPL open source license within the company because doing so "saves engineering time" and because most AGPL projects are of no use to the company."
This is just the way it is -- it's psychology and culture, so deal with it. In contrast, companies very frequently embrace open source code that is licensed under the Apache or BSD licenses, and they keep such projects alive. The extremely popular PostgreSQL database is licensed under an almost-BSD license. MySQL is freely licensed under the GPL, but there are good reasons why people buy a commercial MySQL license (from Oracle) for MySQL. Like RethinkDB, MongoDB is AGPL licensed, but they are happy to sell a different license to companies.

With RethinkDB today, the only option is AGPL. This very strongly discourage use by the only possible group of users and developers that have any chance to keep RethinkDB from death. If this situation is not resolved as soon as possible, I am extremely afraid that it never will be resolved. Ever. If you care about RethinkDB, you should be afraid too. Ignoring the landscape and culture of volunteer open source projects is dangerous.

A Proposal

I don't know who can make the decision to relicense RethinkDB. I don't kow what is going on with investors or who is in control. I am an outsider. Here is a proposal that might provide a way out today:

PROPOSAL: Dear RethinkDB, sell me an Apache (or BSD) license to the RethinkDB source code. Make this the last thing your company sells before it shuts down. Just do it.


Hacker News Discussion

by William Stein ([email protected]) at October 10, 2016 04:03 PM

October 05, 2016

William Stein

SageMath: "it's not research"

The University of Washington (UW) mathematics department has funding for grad students to "travel to conferences". What sort of travel funding?

  • The department has some money available.
  • The UW Graduate school has some money available: They only provide funding for students giving a talk or presenting a poster.
  • The UW GPSS has some money available: contact them directly to apply (they only provide funds for "active conference participation", which I think means giving a talk, presenting a poster, or similar)

One of my two Ph.D. students at UW asked our Grad program director: "I'll be going to Joint Mathematics Meetings (JMM) to help out at the SageMath booth. Is this a thing I can get funding for?"

ANSWER: Travel funds are primarily meant to support research, so although I appreciate people helping out at the SageMath booth, I think that's not the best use of the department's money.

I think this "it's not research" perspective on the value of mathematical software is unfortunate and shortsighted. Moreover, it's especially surprising as the person who wrote the above answer has contributed substantially to the algebraic topology functionality of Sage itself, so he knows exactly what Sage is.

Sigh. Can some blessed person with an NSF grant out there pay for this grad student's travel expenses to help with the Sage booth? Or do I have to use the handful of $10, $50, etc., donations I've got the last few months for this purpose?

by William Stein ([email protected]) at October 05, 2016 01:13 PM

August 22, 2016

Extending Matroid Functionality Google Summer of Code 2016

Overview of what was done

My project has been extending the functionality of SageMath in a matroid direction.
As part of my application, and before the summer officially started, I worked on two tickets: https://trac.sagemath.org/ticket/20290 and https://trac.sagemath.org/ticket/14666. The first was fixing a typo (and learning how to use the interface), and the second one modified the code to find a maximum weighted basis of a matroid so that a user could also see if there was exactly one maximum weighted basis. These are both currently incorporated into official release version of SageMath.

At the beginning of the summer, I was focused on adding certificates to the pre written algorithms is_isomorphic()chordal functionshas_minor(), and has_line_minor(). All of these are closed tickets except the last one, which had a merge conflict. This also enabled me to get a feel for the documentation culture of my organization.

The bulk of my project has been working on implementing An Almost Linear-Time Algorithm for Graph Realization by Robert Bixby and Donald Wagner. This algorithm was written with data structures that didn't exactly match the code base that I was incorporating the function into, so some changes were made there, and some simple (but not necessarily easy) supporting functions were added. There are still some bugs in the code, whose current version can be found here. Much of the rest of this post will be devoted to explaining the data structures that we used for the algorithm. It is aimed mostly at whoever (hopefully future me) is going to finish this function.

We used two new data structures Node, and Decomposition. The decomposition is composed of nodes and relations between them. In particular, it contains a directed tree, where each vertex corresponds to a node. A decomposition also stores information which is useful to the functions that need it. The root of the tree is stored, as are the nodes which contain the first and last verticies of the hypopath along with these verticies. Also stored are integers to makes sure that we don't double name two verticies or two edges the same thing.

A node contains a graph, a parent marker edge, and a parent marker vertex. The latter is one of the vertices of the parent marker edge, and is manipulated so that it is the edge which will end up being included in the path that comes from the hypopath. It also stores an integer T, which depends on the iteration of adding edges, and is stored after being computed.

The flow structure of the main functions is given below. Each function is a decomposition function.


Here is the list of all the functions and the status of each of them. Most of them are supporting functions, with the exception of relink1, typing, relink2, and hypopath from section 4 of the paper, squeeze and update from section 5, and is_graphic from section 6.

Nodes

get_graph(self)
Done
get_parent_marker(self)
Done
get_named_edge(self, f)
Done
get_parent_marker_edge(self)
Done
get_f(self)
Done
set_f(self, int n)
Done
is_polygon(self)
Done
s_path(self, P)
Done
is_cycle(self, P)
Done
_T(self, P, Z=*)
This will correctly give the T value when self is a leaf of the reduced arborescence. It does not correctly compute the T value otherwise.
__relink1(self, Z=*, WQ=*)
Done
__relink2(self, Z=*, WQ=*)
Done
get_T(self)
Done
set_T(self, int T)
Done



CunninghamEdmondsDecomposition

relink1(self, Q, Z=*, WQ=*)
Done
get_D_hat(self, P)
Done
T(self, N, P, T)
This is not done. It needs to be fixed so that it takes into account the types of the children of self.
__typing(self, P, pi)
This is not tested as it relies on T. There are, however, no known deficiencies with the algorithm.
__relink2(Q, Z=*, WQ=*)
Done
__hypopath(self, P)
This is not tested as it relies on __typing. The assigning of u_1 and u_2 needs to be fixed.
__squeeze(self, N, L)
Done
__update(self, P, C)
This is not tested as it relies on __hypopath. It is essentially done, except that the variables u_1, u_2, K_1, and K_2 are not necessarily computed correctly, and U2.4 is not written.
__is_graphic(self)
This is not done. G2 and G3 need to be written, and it needs to be tested. This cannot happen until the rest of the problems are fixed.
merge_with_parent(self, N, N_vertex=*, P_vertex=*)
This is done, but it doesn't use the f is N_vertex and P_vertex are undefined. This should probably be changed.
merge_branch(self, N, P)
This is written, but in order to insure that the intersection of P with this graph is always a path if possible, P should be replaced with P_0, and the parent markers of children that intersect P should be added to P_0 initially, and removed, in turn, when that child is merged with N.
__add_cycle(self, cycle)
Done
get_arborescence(self)
Done
get_nodes(self)
Done
get_root(self)
Done
__get_pi(self)
This is done, but it should be changed so that it can take a sub tree of self.arborescence as an input, and give pi on the reduced decomposition.
branch(self, N)
Done
get_parent(self, N)
 Done

by [email protected] (Tara) at August 22, 2016 02:18 PM

Lauren Devitt

Final GSOC Report

Over the summer I have been coding for SAGE as part of Google Summer of Code 2016.  I feel this opportunity has strengthened both my skills as a coder and as a mathematician. SAGE is an open source math software system, used to do calculations.  SAGE is made up of code available for free public use and modification. It is a collaborative effort where people from around the world can add new code, update old code, and share the changes that they make. Open source coding is a labor of

August 22, 2016 12:00 AM

August 12, 2016

William Stein

Jupyter: "take the domain name down immediately"

The Jupyter notebook is an open source BSD-licensed browser-based code execution environment, inspired by my early work on the Sage Notebook (which we launched in 2007), which was in turn inspired heavily by Mathematica notebooks and Google docs. Jupyter used to be called IPython.

SageMathCloud is an open source web-based environment for using Sage worksheets, terminals, LaTeX documents, course management, and Jupyter notebooks. I've put much hard work into making it so that multiple people can simultaneously edit Jupyter notebooks in SageMathCloud, and the history of all changes are recorded and browsable via a slider.

Many people have written to me asking for there to be a modified version of SageMathCloud, which is oriented around Jupyter notebooks instead of Sage worksheets. So the default file type is Jupyter notebooks, the default kernel doesn't involve the extra heft of Sage, etc., and the domain name involves Jupyter instead of "sagemath". Some people are disuased from using SageMathCloud for Jupyter notebooks because of the "SageMath" name.

Dozens of web applications (including SageMathCloud) use the word "Jupyter" in various places. However, I was unsure about using "jupyter" in a domain name. I found this github issue and requested clarification 6 weeks ago. We've had some back and forth, but they recently made it clear that it would be at least a month until any decision would be considered, since they are too busy with other things. In the meantime, I rented jupytercloud.com, which has a nice ring to it, as the planet Jupiter has clouds. Yesterday, I made jupytercloud.com point to cloud.sagemath.com to see what it would "feel like" and Tim Clemans started experimenting with customizing the page based on the domain name that the client sees. I did not mention jupytercloud.com publicly anywhere, and there were no links to it.

Today I received this message:

    William,

I'm writing this representing the Jupyter project leadership
and steering council. It has recently come to the Jupyter
Steering Council's attention that the domain jupytercloud.com
points to SageMathCloud. Do you own that domain? If so,
we ask that you take the domain name down immediately, as
it uses the Jupyter name.
I of course immediately complied. It is well within their rights to dictate how their name is used, and I am obsessive about scrupulously doing everything I can to respect people's intellectual property; with Sage we have put huge amounts of effort into honoring both the letter and spirit of copyright statements on open source software.

I'm writing this because it's unclear to me what people really want, and I have no idea what to do here.

1. Do you want something built on the same technology as SageMathCloud, but much more focused on Jupyter notebooks?

2. Does the name of the site matter to you?

3. What model should the Jupyter project use for their trademark? Something like Python? like Git?Like Linux?  Like Firefox?  Like the email program PINE?  Something else entirely?

4. Should I be worried about using Jupyter at all anywhere? E.g., in this blog post? As the default notebook for the SageMath project?

I appreciate any feedback.

Hacker News Discussion

UPDATE (Aug 12, 2016): The official decision is that I cannot use the domain jupytercloud.com.   They did say I can use jupyter.sagemath.com or sagemath.com/jupyter.   Needless to say, I'm disappointed, but I fully respect their (very foolish, IMHO) decision.


by William Stein ([email protected]) at August 12, 2016 10:46 AM

July 27, 2016

Lauren Devitt

Poetry By Number

In the 1960’s French writer and poet Raymond Queneau became the most prolific writers of our time by writing over one hundred thousand billion poems.  If you wanted to read all of these poems it would only take you 200 million years of reading 24 hours a day at a rate of about one poem per minute.  So how could Queneau write so many poems in far less time than it would take to read them all? The truth is he didn’t actually write that many individual poems. What he did was write ten 14-line

July 27, 2016 12:00 AM

July 22, 2016

William Stein

DataDog's pricing: don't make the same mistake I made

I stupidly made a mistake recently by choosing to use DataDog for monitoring the infrastructure for my startup (SageMathCloud).

I got bit by their pricing UI design that looks similar to many other sites, but is different in a way that caused me to spend far more money than I expected.

I'm writing this post so that you won't make the same mistake I did.  As a product, DataDog is of course a lot of hard work to create, and they can try to charge whatever they want. However, my problem is that what they are going to charge was confusing and misleading to me.

I wanted to see some nice web-based data about my new autoscaled Kubernetes cluster, so I looked around at options. DataDog looked like a new and awesomely-priced service for seeing live logging. And when I looked (not carefully enough) at the pricing, it looked like only $15/month to monitor a bunch of machines. I'm naive about the cost of cloud monitoring -- I've been using Stackdriver on Google cloud platform for years, which is completely free (for now, though that will change), and I've also used self hosted open solutions, and some quite nice solutions I've written myself. So my expectations were way out of whack.

Ever busy, I signed up for the "$15/month plan":


One of the people on my team spent a little time and installed datadog on all the VM's in our cluster, and also made DataDog automatically start running on any nodes in our Kubernetes cluster. That's a lot of machines.

Today I got the first monthly bill, which is for the month that just happened. The cost was $639.19 USD charged to my credit card. I was really confused for a while, wondering if I had bought a year subscription.



After a while I realized that the cost is per host! When I looked at the pricing page the first time, I had just saw in big letters "$15", and "$18 month-to-month" and "up to 500 hosts". I completely missed the "Per Host" line, because I was so naive that I didn't think the price could possibly be that high.

I tried immediately to delete my credit card and cancel my plan, but the "Remove Card" button is greyed out, and it says you can "modify your subscription by contacting us at [email protected]":



So I wrote to [email protected]:

Dear Datadog,

Everybody on my team was completely mislead by your
horrible pricing description.

Please cancel the subscription for wstein immediately
and remove my credit card from your system.

This is the first time I've wasted this much money
by being misled by a website in my life.

I'm also very unhappy that I can't delete my credit
card or cancel my subscription via your website. It's
like one more stripe API call to remove the credit card
(I know -- I implemented this same feature for my site).


And they responded:

Thanks for reaching out. If you'd like to cancel your
Datadog subscription, you're able to do so by going into
the platform under 'Plan and Usage' and choose the option
downgrade to 'Lite', that will insure your credit card
will not be charged in the future. Please be sure to
reduce your host count down to the (5) allowed under
the 'Lite' plan - those are the maximum allowed for
the free plan.

Also, please note you'll be charged for the hosts
monitored through this month. Please take a look at
our billing FAQ.


They were right -- I was able to uninstall the daemons, downgrade to Lite, remove my card, etc. all through the website without manual intervention.

When people have been confused with billing for my site, I have apologized, immediately refunded their money, and opened a ticket to make the UI clearer.  DataDog didn't do any of that.

I wish DataDog would at least clearly state that when you use their service you are potentially on the hook for an arbitrarily large charge for any month. Yes, if they had made that clear, they wouldn't have had me as a customer, so they are not incentivized to do so.

A fool and their money are soon parted. I hope this post reduces the chances you'll be a fool like me.  If you chose to use DataDog, and their monitoring tools are very impressive, I hope you'll be aware of the cost.


ADDED:

On Hacker News somebody asked: "How could their pricing page be clearer? It says per host in fairly large letters underneath it. I'm asking because I will be designing a similar page soon (that's also billed per host) and I'd like to avoid the same mistakes."  My answer:

[EDIT: This pricing page by the top poster in this thread is way better than I suggest below -- https://www.serverdensity.com/pricing/]

1. VERY clearly state that when you sign up for the service, then you are on the hook for up to $18*500 = $9000 + tax in charges for any month. Even Google compute engine (and Amazon) don't create such a trap, and have a clear explicit quota increase process.
2. Instead of "HUGE $15" newline "(small light) per host", put "HUGE $18 per host" all on the same line. It would easily fit. I don't even know how the $15/host datadog discount could ever really work, given that the number of hosts might constantly change and there is no prepayment.
3. Inform users clearly in the UI at any time how much they are going to owe for that month (so far), rather than surprising them at the end. Again, Google Cloud Platform has a very clear running total in their billing section, and any time you create a new VM it gives the exact amount that VM will cost per month.
4. If one works with a team, 3 is especially important. The reason that I had monitors on 50+ machines is that another person working on the project, who never looked at pricing or anything, just thought -- he I'll just set this up everywhere. He had no idea there was a per-machine fee.

by William Stein ([email protected]) at July 22, 2016 02:17 PM

June 27, 2016

Extending Matroid Functionality Google Summer of Code 2016

Midterm ish

My summer of code is broken up into several projects. There were a lot of small ones, a couple medium ones, and one large one. Right now, I'm in the midst of working on the large project. Basically, we want to feed Sage a collection of subsets of an edge set E, and have Sage tell us if there is a graph that has cycles which correspond to the subsets of E, and if so, to give a corresponding. This boils down to asking if a matroid is graphic, and asking for a graph that realizes the matroid.

For instance, if we give have E = {1, 2, 3, 4}, and our collection of sets is any three element subset of E, then we can't get an appropriate graph. To see this, we start constructing a graph. Our first cycle is {1, 2, 3}, There is only one graph on three elements that has this cycle, namely a triangle. To add the edge 4, we need to have a cycle {1, 2, 4}. But this means that we have to add 4 in parallel to the edge 3. This is a problem, because then {1, 3, 4}, in particular, is not a cycle of our graph.

This example illustrates a key idea of the algorithm. The set {1, 2} is a maximal set that is not contained in a cylce, so we skipped over those elements, and started with 3. We then added 3 and any needed elements of {1, 2} to our partial graph. And we kept adding elements till we either had a problem, or till we added all of the elements.

In our case, we didn't get so complicated of a graph that we had a choice about which graph to use for our partial graph. In general, this is not the case. It would be troublesome to check if we could add the new element to every graphs that realizes the already added elements, so we use a decomposition made possible by Whitney's 2 isomorphism theorem to check all of the graphs options at once. This of course makes the code more complicated. The algorithm that we are following comes from a paper by Ronald Bixby and Donald Wagner.

The tricky part, so far, has been trying to get information in and out of graphs. graph theorists care a lot about the vertices of a graph and much less about the edges of the graph. That is, they store their edges as a list of the two vertices that they are incident with, and a possible label. matroid theorists, however, care a lot more about the edges of a graph. This is true in general, and is true in particular for this project.

by [email protected] (Tara) at June 27, 2016 05:41 PM

June 23, 2016

Lauren Devitt

GSOC Midterm Update

One of the great joys in life that not all will be able to experience is when your code builds. When you have fixed all the syntax errors you build it, you test it, and you see ---------------------------------------------------------------------- All tests passed! ---------------------------------------------------------------------- Having to wait for your code to build is agonizing. You pray to some God, magic force, or Steve Jobs to help you in this time of need. Then it works and you feel

June 23, 2016 12:00 AM

June 09, 2016

Lauren Devitt

GSOC Week 2 Update

Two weeks into GSOC 2016 I’m so grateful that this is how I get to spend my summer. Using my coding muscles has made me stronger and more confident with my code. I am ready to create my first ticket in SAGE relating to GSOC. Sage uses Trac, which is an open source project managing software. It allows SAGE to track what people are currently working on for SAGE. It does this by are giving a ticket number to each piece of functionality pushed to the server. Peers, who pull them from the server,

June 09, 2016 12:00 AM

June 03, 2016

Extending Matroid Functionality Google Summer of Code 2016

First Week or so

Before coding started, I spent some time on code academy getting more familiar with the syntax of Python. I was impressed with the setup that they had (I would recommend it to my mom), and it helped me to learn python in a systematic way.

Since the 23rd I've been working on adding certificated (proof that we gave the right answer to a yes-no question) to some of the functions in the matroid part of Sage. For the first two days, I spent a lot of time trying to get Sage to compile. For a while, the problem was an error in a new release, and then I had some type of trouble on my end. I've also spent a good amount of time figuring out the ins and outs of documentation practices.

by [email protected] (Tara) at June 03, 2016 12:30 AM

May 21, 2016

Lauren Devitt

GSOC 2016

What I love about programing is it is akin to solving a logic puzzle. You have all the pieces you need to solve it, you're allowed to search the internet for assistance, but the internet will not give you the answer, you still have to find it on you own. If you have misread one letter or decoded one word wrong, you will not find the answer. You can spend twice as much time trying to solve the problem than actually solving the puzzle. That is why in coding, as with logic puzzles,

May 21, 2016 12:00 AM

May 20, 2016

Lauren Devitt

For love of numbers...

My love of mathematics started with a love of numbers. I enjoyed finding all possible ways I could add, subtract, and multiply different numbers in order to find a specific number, say twelve.  Twelve was my favorite number; I loved twelve. We become attached to numbers that have a deeper meaning to us; numbers that make us feel, make us remember. Sesame Street had the “Pinball Song”; this song consisted of counting to twelve with a catchy jingle every child could remember.

May 20, 2016 12:00 AM

May 09, 2016

Extending Matroid Functionality Google Summer of Code 2016

Getting Started

I first heard about Google Summer of Code a little over a year ago. It was something that I wanted to do for several reasons. I only had a chance to take a couple of programing classes in undergrad. (I didn't realized that I liked it till part way through my Junior year.) Since then, I've wanted to grow the length and complexity of projects that I was capable of successfully working on. Secondly, I like the idea of open source resources, because its free, and that lets poor college students use cool resources.

My project is building and expanding tools in Sage to be used by people studying matroid theory. A matroid is a notion of independence that generalizes the independence structure that is found in vector spaces and that comes from looking at cycleless subgraphs of graphs. Sage already has a lot of tools that let people work with matroids, mostly created by Stefan van Zwam and Rudi Pendavingh. My project focuses on a small collection of new tools.

I'll be working with Stefan and Michael on this project.

by [email protected] (Tara) at May 09, 2016 05:29 PM

March 10, 2016

William Stein

Open source is now ready to compete with Mathematica for use in the classroom



When I think about what makes SageMath different, one of the most fundamental things is that it was created by people who use it every day.  It was created by people doing research math, by people teaching math at universities, and by computer programmers and engineers using it for research.  It was created by people who really understand computational problems because we live them.  We understand the needs of math research, teaching courses, and managing an open source project that users can contribute to and customize to work for their own unique needs.

The tools we were using, like Mathematica, are clunky, very expensive, and just don't do everything we need.  And worst of all, they are closed source software, meaning that you can't even see how they work, and can't modify them to do what you really need.  For teaching math, professors get bogged down scheduling computer labs and arranging for their students to buy and install expensive software.

So I started SageMath as an open source project at Harvard in 2004, to solve the problem that other math software is expensive, closed source, and limited in functionality, and to create a powerful tool for the students in my classes.  It wasn't a project that was intended initially as something to be used by hundred of thousands of people.  But as I got into the project and as more professors and students started contributing to the project, I could clearly see that these weren't just problems that pissed me off, they were problems that made everyone angry.

The scope of SageMath rapidly expanded.  Our mission evolved to create a free open source serious competitor to Mathematica and similar closed software that the mathematics community was collective spending hundreds of millions of dollars on every year. After a decade of work by over 500 contributors, we made huge progress.

But installing SageMath was more difficult than ever.  It was at that point that I decided I needed to do something so that this groundbreaking software that people desperately needed could be shared with the world.

So I created SageMathCloud, which is an extremely powerful web-based collaborative way for people to easily use SageMath and other open source software such as LaTeX, R, and Jupyter notebooks easily in their teaching  and research.   I created SageMathCloud based on nearly two decades of experience using math software in the classroom and online, at Harvard, UC San Diego, and University of Washington.

SageMathCloud is commercial grade, hosted in Google's cloud, and very large classes are using it heavily right now.  It solves the installation problem by avoiding it altogether.  It is entirely open source.

Open source is now ready to directly compete with Mathematica for use in the classroom.  They told us we could never make something good enough for mass adoption, but we have made something even better.  For the first time, we're making it possible for you to easily use Python and R in your teaching instead of Mathematica; these are industry standard mainstream open source programming languages with strong support from Google, Microsoft and other industry leaders.   For the first time, we're making it possible for you to collaborate in real time and manage your course online using the same cutting edge software used by elite mathematicians at the best universities in the world.

A huge community in academia and in industry are all working together to make open source math software better at a breathtaking pace, and the traditional closed development model just can't keep up.

by William Stein ([email protected]) at March 10, 2016 08:02 AM

March 03, 2016

Liang Ze

Noncommutative Algebras in Sage

In this post, I’ll demonstrate 3 ways to define non-commutative rings in Sage. They’re essentially different ways of expressing the non-commutative relations in the ring:

  1. Via g_algebra: define the relations directly
  2. Via NCPolynomialRing_plural: define a pair of structural matrices
  3. Via a quotient of a letterplace ring: define the ideal generated by the relations (only works for homogeneous relations)

As far as I know, all 3 methods rely on Sage’s interface with Singular and its non-commutative extension Plural.

In addition to all the documentation linked above, I also relied heavily on Greuel and Pfister’s A Singular Introduction to Commutative Algebra. Despite the title, it does have a pretty substantial section (1.9) devoted to non-commutative $G$-algebras.

$U(\mathfrak{sl}_2)$ and its homogenization

The running example throughout this post will be the universal enveloping algebra $U(\mathfrak{sl}_2)$ over $\mathbb{Q}$.

We’ll define this to be the (non-commutative) $\mathbb{Q}$-algebra $U$ with generators $e,f,h$ subject to the relations

If we set $e,f,h$ to have degree 1, these relations are not homogeneous. Their left-hand sides only have degree 2 terms, while their right-hand sides have degree 1 terms as well. This is fine with the first two methods, but won’t work for method 3 (which requires homogeneous relations).

To demonstrate the third method, we’ll define the $\mathbb{Q}$-algebra $H$ with generators $e,f,h,t$ subject to the homogeneous relations

We can obtain $U$ both as a quotient and a localization of $H$:

$G$-algebras

Using the g_algebra method of Sage’s FreeAlgebra class, we can simply plug our noncommutative relations in, and get our non-commutative ring. This is about as easy as it gets:

Let’s unravel what’s going on here.

Monomial orderings and PBW basis

Most algorithms for commutative and non-commutative rings require an ordering on the generators. In our case, let’s use the ordering

This is implicitly stated in our code: we wrote F.<e,f,h> instead of F.<h,e,f>, for example.

A standard word is a monomial of the form

In the polynomial ring $\mathbb{Q}[e,f,h]$, every monomial can be expressed in this form, so the set of standard words forms a $\mathbb{Q}$-basis for $\mathbb{Q}[e,f,h]$.

In a non-commutative ring, whether or not the standard words form a basis depends on what relations we have. Such a basis, if it exists, is called a PBW basis.

The free algebra $F = \mathbb{Q}\langle e,f,h\rangle$ has no relations, so does not have a PBW basis. Fortunately, our algebra $U$ does have a PBW basis.

This means that we can always express a non-standard monomial (e.g. $fe$) as a sum of standard monomials (e.g. $ef - h$). The non-commutative relations that define $U$ can thus be thought of as an algorithm for turning non-standard words into sums of standard words.

To do this in Sage, we define a dictionary whose keys are non-standard words and values are the standard words they become.

In the above example, our dictionary was short enough to fit into one line, but we could also define a dictionary separately and pass it into g_algebra:

It’s very important that the keys are non-standard words and the values are sums of standard words. Mathematically, the relation $fe = ef - h$ is the same as $ef = fe + h$, but if we replace f*e : e*f - h with e*f : f*e + h in the code, we’ll get an error (try it!).

What are $G$-algebras?

The reason why $U$ has a PBW basis is because it is a $G$-algebra. Briefly, $G$-algebras are algebras whose relations satisfy certain non-degeneracy conditions that make the algebra nice to work with.

For a full definition of $G$-algebras, refer to A Singular Introduction to Commutative Algebra or the Plural manual.

If $A$ is a $G$-algebra, then it has a PBW basis, is left and right Noetherian, and is an integral domain. More importantly (for this site at least!), it means that we can define $A$ in Singular/Plural, and hence in Sage.

Structural matrices for a $G$-algebra

Another way of writing our non-commutative relations is

where $ * $ denotes element-wise multiplication (so there isn’t any linear algebra going on here; we’re just using matrices to organize the information). Let $N,C,S,D$ be the matrices above, in that order, so that $N = C*S + D$.

If we let $x_1 = e, x_2 = f, x_3 = h$ (so that $x_i \leq x_j$ if $i \leq j$) then for $i < j$

In other words, $N$ contains the non-standard words that we’re trying to express in terms of the standard words in $S$.

The matrices $C$ and $D$ are called the structural matrices of the $G$-algebra, and their entries are such that our relations may be written

with zeros everywhere else ($i \geq j$). If $C = D = 0$, the resulting algebra will be commutative.

We can use the structural matrices $C$ and $D$ to define our algebra via Sage’s NCPolynomialRing_plural function (note that Python uses zero-indexing for matrices):

Note that R is a commutative polynomial ring. In fact, up till the point where we call NCPolynomialRing_plural, even the variables e,f,h are treated as commutative variables.

This method of defining $U$ is considerably longer and more prone to mistakes than using g_algebra. As stated in the documentation, this is not intended for use! I’m including it here because this is essentially how one would go about defining a $G$-algebra in Singular. In fact, the Sage method g_algebra calls NCPolynomialRing_plural, which in turn calls Singular.

Quotients of letterplace rings

Our final method for defining non-commutative rings makes use of Sage’s implementation of Singular’s letterplace rings.

As mentioned at the start of this post, this method requires the relations to be homogeneous, so we’ll work with $H$ instead of $U$.

Let $\mathbb{Q}\langle e,f,h,t \rangle$ be the free algebra on 4 variables. Consider the two-sided ideal $I$ generated by the relations for $H$:

Then

This can be expressed Sage-ly:

The expression F*I*F is the two-sided ideal generated by elements in the list I.

Although $U$ cannot be defined using this method, $H$ can be defined using all three methods. As a (fun?) exercise, try defining $H$ using the other two methods.

Difficulties

These methods can be used to define many non-commutative algebras such as the Weyl algebra and various enveloping algebras of Lie algebras. One can also define these algebras over fields other than $\mathbb{Q}$, such as $\mathbb{C}$ or $\mathbb{F}_p$.

However, we cannot define algebras over $\mathbb{Q}(q)$, the fraction field of $\mathbb{Q}[q]$:

This is a problem if we want to define rings with relations such as

Such relations occur frequently when studying quantum groups, for example.

This is suprising, because one can easily define $\mathbb{Q}(q)$ and non-commutative $\mathbb{Q}(q)$-algebras in Singular/Plural, which is what Sage is using. It seems that the problem is in Sage’s wrapper for Singular/Plural, because Sage can’t even pass the ring $\mathbb{Q}(q)$ to Singular.

There’s a trac ticket for this problem, but until it gets resolved, we’ll just have to define such rings directly in Singular/Plural. Thanks to the amazing capabilities of the Sage Cell Server, we’ll do this in the next post!

March 03, 2016 12:00 AM

February 25, 2016

William Stein

"If you were new faculty, would you start something like SageMathCloud sooner?"

I was recently asked by a young academic: "If you were a new faculty member again, would you start something like SageMathCloud sooner or simply leave for industry?" The academic goes on to say "I am increasingly frustrated by continual evidence that it is more valuable to publish a litany of computational papers with no source code than to do the thankless task of developing a niche open source library; deep mathematical software is not appreciated by either mathematicians or the public."

I wanted to answer that "things have gotten better" since back in 2000 when I started as an academic who does computation. Unfortunately, I think they have gotten worse. I do not understand why. In fact, this evening I just received the most recent in a long string of rejections by the NSF.

Regarding a company versus taking a job in industry, for me personally there is no point in starting a company unless you have a goal that can only be accomplished via a company, since building a business from scratch is extremely hard and has little to do with math or research. I do have such a goal: "create a viable open source alternative to Mathematica, etc...". I was very clearly told by Michael Monagan (co-founder of Maplesoft) in 2006 that this goal could not be accomplished in academia, and I spent the last 10 years trying to prove him wrong.

On the other hand, leaving for a job in industry means that your focus will switch from "pure" research to solving concrete problems that make products better for customers. That said, many of the mathematicians who work on open source math software do so because they care so much about making the experience of using math software much better for the math community. What often drives Sage developers is exactly the sort of passionate care for "consumer focus" and products that also makes one successful in industry. I'm sure you know exactly what I mean, since it probably partly motivates your work. It is sad that the math community turns its back on such people. If the community were to systematically embrace them, instead of losing all these $300K+/year engineers to mathematics entirely -- which is exactly what we do constantly -- the experience of doing mathematics could be massively improved into the future. But that is not what the community has chosen to do. We are shooting ourselves in the foot.

Now that I have seen how academia works from the inside over 15 years I'm starting to understand a little why these things change very slowly, if ever. In the mathematics department I'm at, there are a small handful of research areas in pure math, and due to how hiring works (voting system, culture, etc.) we have spent the last 10 years hiring in those areas little by little (to replace people who die/retire/leave). I imagine most mathematics departments are very similar. "Open source software" is not one of those traditional areas. Nobody will win a Fields Medal in it.

Overall, the mathematical community does not value open source mathematical software in proportion to its value, and doesn't understand its importance to mathematical research and education. I would like to say that things have got a lot better over the last decade, but I don't think they have. My personal experience is that much of the "next generation" of mathematicians who would have changed how the math community approaches open source software are now in industry, or soon will be, and hence they have no impact on academic mathematical culture. Every one of my Ph.D. students are now at Google/Facebook/etc.

We as a community overall would be better off if, when considering how we build departments, we put "mathematical software writers" on an equal footing with "algebraic geometers". We should systematically consider quality open source software contributions on a potentially equal footing with publications in journals.

To answer the original question, YES, knowing what I know now, I really wish I had started something like SageMathCloud sooner. In fact, here's the previously private discussion from eight years ago when I almost did.

--

- There is a community generated followup ...

by William Stein ([email protected]) at February 25, 2016 03:14 PM

February 24, 2016

William Stein

Elliptic curves: Magma versus Sage

Elliptic Curves

Elliptic curves are certain types of nonsingular plane cubic curves, e.g., y^2 = x^3 + ax +b, which are central to both number theory and cryptography (e.g., they are used to compute the hash in bitcoin).


Magma and Sage

If you want to do a wide range of explicit computations with elliptic curves, for research purposes, you will very likely use SageMath or Magma. If you're really serious, you'll use both.

Both Sage and Magma are far ahead of all other software (e.g., Mathematica, Maple and Matlab) for elliptic curves.

A Little History

When I started contributing to Magma in 1999, I remember that Magma was way, way behind Pari. I remember having lunch with John Cannon (founder of Magma), and telling him I would no longer have to use Pari if only Magma would have dramatically faster code for computing point counts on elliptic curves.

A few years later, John wisely hired Mark Watkins to work fulltime on Magma, and Mark has been working there for over a decade. Mark is definitely one of the top people in the world at implementing (and using) computational number theory algorithms, and he's ensured that Magma can do a lot. Some of that "do a lot" means catching up with (and surpassing!) what was in Pari and Sage for a long time (e.g., point counting, p-adic L-functions, etc.)

However, in addition, many people have visited Sydney and added extremely deep functionality for doing higher descents to Magma, which is not available in any open source software. Search for Magma in this paper to see how, even today, there seems to be no open source way to compute the rank of the curve y2 = x3 + 169304x + 25788938.  (The rank is 0.)

Two Codebases

There are several elliptic curves algorithms available only in Magma (e.g., higher descents) ... and some available only in Sage (L-function rank bounds, some overconvergent modular symbols, zeros of L-functions, images of Galois representations). I could be wrong about functionality not being in Magma, since almost anything can get implemented in a year...

The code bases are almost completely separate, which is a very good thing. Any time something gets implemented in one, it gets (or should get) tested via a big run on elliptic curves up to some bound in the other. This sometimes results in bugs being found. I remember refereeing the "integral points" code in Sage by running it against all curves up to some bound and comparing to what Magma output, and getting many discrepancies, which showed that there were bugs in both Sage and Magma.
Thus we would be way better off if Sage could do everything Magma does (and vice versa).



by William Stein ([email protected]) at February 24, 2016 12:07 PM

February 18, 2016

Sébastien Labbé

unsupported operand parent for *, Matrix over number field, vector over symbolic ring

Yesterday I received this email (in french):

Salut,
avec Thomas on a une question bête:

K.<x>=NumberField(x*x-x-1)

J'aimerais multiplier une matrice avec des coefficients en x par un vecteur
contenant des variables a et b.  Il dit "unsupported operand parent for *,
Matrix over number field, vector over symbolic ring"

Est ce grave ?

Here is my answer. Indeed, in Sage, symbolic variables can't multiply with elements in an Number Field in x:

sage: x = var('x')
sage: K.<x> = NumberField(x*x-x-1)
sage: a = var('a')
sage: a*x
Traceback (most recent call last)
...
TypeError: unsupported operand parent(s) for '*': 'Symbolic Ring' and
'Number Field in x with defining polynomial x^2 - x - 1'

But, we can define a polynomial ring with variables in a,b and coefficients in the NumberField. Then, we are able to multiply a with x:

sage: x = var('x')
sage: K.<x> = NumberField(x*x-x-1)
sage: K
Number Field in x with defining polynomial x^2 - x - 1
sage: R.<a,b> = K['a','b']
sage: R
Multivariate Polynomial Ring in a, b over Number Field in x with
defining polynomial x^2 - x - 1
sage: a*x
(x)*a

With two square brackets, we obtain powers series:

sage: R.<a,b> = K[['a','b']]
sage: R
Multivariate Power Series Ring in a, b over Number Field in x with
defining polynomial x^2 - x - 1
sage: a*x*b
(x)*a*b

It works with matrices:

sage: MS = MatrixSpace(R,2,2)
sage: MS
Full MatrixSpace of 2 by 2 dense matrices over Multivariate Power
Series Ring in a, b over Number Field in x with defining polynomial
x^2 - x - 1
sage: MS([0,a,b,x])
[  0   a]
[  b (x)]
sage: m1 = MS([0,a,b,x])
sage: m2 = MS([0,a+x,b*b+x,x*x])
sage: m1 + m2 * m1
[              (x)*b + a*b       (x + 1) + (x + 1)*a]
[                (x + 2)*b (3*x + 1) + (x)*a + a*b^2]

by Sébastien Labbé at February 18, 2016 10:17 AM

February 17, 2016

Liang Ze

The Weyl Algebra and $\mathfrak{sl}_2$

I’ve been away from this blog for quite a while - almost a year, in fact! My excuses are my wedding and the prelims (a.k.a. quals), as well as all the preparation that had to go into them (although, to be honest, those things only occupied me till September last year!).

Looking back at my previous posts, I’ve realized that in attempting to teach both math and code, I probably ended up doing neither. This is really not the best place to learn representation theory (for example) - there are better books and blogs out there. Also, most of the code that I wrote to illustrate those posts feels contrived, and neither highlights Sage’s strengths nor reflects how I normally use Sage for my assignments and projects.

I’ve thus decided to write shorter posts with code that I actually use (on SageMathCloud), along with some explanations of the code. Lately, I’ve been writing code for non-commutative algebra and combinatorics, so today I’ll start with a simple example of a non-commutative algebra.

The Weyl Algebra

The $1$-dim. Weyl algebra is the (non-commutative) algebra generated by $x, \partial_x$ subject to the relations

If we treat $x$ as “multiplication by $x$” and $\partial_x$ as “differentiation w.r.t. $x$”, this relation is really just an application of the chain rule:

We can generalize to higher dimensions: the $n$-dim. Weyl algebra is the algebra generated by $x_1,\dots,x_n,\partial_{x_1},\dots,\partial_{x_n}$ quotiented by the relations that arise from treating them as the obvious operators on $\mathbb{F}[x_1,\dots,x_n]$.

Weyl algebras in Sage

It’s easy to define the Weyl algebra in Sage:

Calling inject_variables allows us to use the operators x,y,z,dx,dy,dz in subsequent code (where dx denotes $\partial_x$, etc).

One can do rather complicated computations:

By default, Sage chooses to represent monomials with x,y,z in front of dx,dy,dz:

Keep in mind that x does not refer to the polynomial $x \in \mathbb{F}[x]$, so one should not expect dx*x to be 1.

(For some reason show does not give the right output. Try show(x) or show(x*dx), for example.)

Representations of $\mathfrak{sl}_2$

It turns out that the $1$-dim. Weyl algebra gives a representation of $\mathfrak{sl}_2(\mathbb{F})$.

The Lie algebra $\mathfrak{sl}_2(\mathbb{F})$ is generated by $E,F,H$ subject to the relations

Define the following elements of the $1$-dim. Weyl algebra:

We can use Sage to quickly verify that these elements indeed satisfy the relations for $\mathfrak{sl}_2$ (using the commutator as the Lie bracket i.e. $[A,B] = AB - BA$):

Working over $\mathbb{C}$, this action of $\mathfrak{sl}_2(\mathbb{C})$ makes $\mathbb{C}[x]$ a Verma module of highest weight $0$.

In fact, we can make $\mathbb{C}[x]$ a Verma module of highest weight $c$ for any $c \in \mathbb{C}$ by using:

We verify this again in Sage:

In subsequent posts, I’ll talk more about defining other non-commutative algebras in Sage and Singular.

February 17, 2016 12:00 AM

January 15, 2016

William Stein

Thinking of using SageMathCloud in a college course?

SageMathCloud course subscriptions

"We are  college instructors of the calculus sequence and ODE’s.  If the college were to purchase one of the upgrades for us as we use Sage with our students, who gets the benefits of the upgrade?  Is is the individual students that are in an instructor’s Sage classroom or is it the  collaborators on an instructor’s project?"

If you were to purchase just the $7/month plan and apply the upgrades to *one* single project, then all collaborators on that one project would benefit from those upgrades while using that project.

If you were to purchase a course plan for say $399/semester, then you could apply the upgrades (network access and members only hosting) to 70 projects that you might create for a course.   When you create a course by clicking +New, then "Manage a Course", then add students, each student has their own project created automatically.  All instructors (anybody who is a collaborator on the project where you clicked "Manage a course") is also added to the student's project. In course settings you can easily apply the upgrades you purchase to all projects in the course.  

Also I'm currently working on a new feature where instructors may choose to require all students in their course to pay for the upgrade themselves.  There's a one time $9/course fee paid by the student and that's it.  At some colleges (in some places) this is ideal, and at other places it's not an option at all.   I anticipate releasing this very soon.





Getting started with SageMathCloud courses


You can fully use the SMC course functionality without paying anything in order to get familiar with it and test it out.  The main benefit of paying is that you get network access and all projects get moved to members only servers, which are much more robust; also, we greatly prioritize support for paying customers.   

This blog post is an overview of using SMC courses:

  http://www.beezers.org/blog/bb/2015/09/grading-in-sagemathcloud/

This has some screenshots and the second half is about courses:

  http://blog.ouseful.info/2015/11/24/course-management-and-collaborative-jupyter-notebooks-via-sagemathcloud/

Here are some video tutorials made by an instructor that used SMC with a large class in Iceland recently:

  https://www.youtube.com/watch?v=dgTi11ZS3fQ
  https://www.youtube.com/watch?v=nkSdOVE2W0A
  https://www.youtube.com/watch?v=0qrhZQ4rjjg

Note that the above videos show the basics of courses, then talk specifically about automated grading of Jupyter notebooks.  That might not be at all what you want to do -- many math courses use Sage worksheets, and probably don't automate the grading yet.

Regarding using Sage itself for teaching your courses, check out the free pdf book to "Sage for Undergraduates" here, which the American Mathematical Society just published (there is also a very nice print version for about $23):

   http://www.gregorybard.com/SAGE.html

by William Stein ([email protected]) at January 15, 2016 09:14 AM

January 08, 2016

William Stein

Mathematics Graduate School: preparation for non-academic employment

This is about my personal experience as a mathematics professor whose students all have non-academic jobs that they love. This is in preparation for a panel at the Joint Mathematics Meetings in Seattle.

My students and industry
My graduated Ph.D. students:
  • 3 at Google
  • 1 at Facebook
  • 1 at CCR
My graduating student (Hao Chen):
  • Applying for many postdocs
  • But just did summer internship at Microsoft Research with Kristin. (I’ve had four students do summer internships with Kristin)
All my students:
  • Have done a lot of Software development, maybe having little to do with math, e.g., “developing the Cython compiler”, “transition the entire Sage project to git”, etc.
  • Did a thesis squarely in number theory, with significant theoretical content.
  • Guilt (or guilty pleasure?) spending time on some programming tasks instead of doing what they are “supposed” to do as math grad students.

Me: academia and industry

  • Math Ph.D. from Berkeley in 2000; many students of my advisor (Lenstra) went to work at CCR after graduating…
  • Academia: I’m a tenured math professor (since 2005) – number theory.
  • Industry: I founded a Delaware C Corp (SageMath, Inc.) one year ago to “commercialize Sage” due to VERY intense frustration trying to get grant funding for Sage development. Things have got so bad, with so many painful stupid missed opportunities over so many years, that I’ve given up on academia as a place to build Sage.
Reality check: Academia values basic research, not products. Industry builds concrete valuable products. Not understanding this is a recipe for pain (at least it has been for me).

Advice for students from students

Robert Miller (Google)

My student Robert Miller’s post on Facebook yesterday: “I LOVE MY JOB”. Why: “Today I gave the first talk in a seminar I organized to discuss this result: ‘Graph Isomorphism in Quasipolynomial Time’. Dozens of people showed up, it was awesome!”
Background: When he was my number theory student, working on elliptic curves, he gave a talk about graph theory in Sage at a Sage Days (at IPAM). His interest there was mainly in helping an undergrad (Emily Kirkman) with a Sage dev project I hired her to work on. David Harvey asked: “what’s so hard about implementing graph isomorphism”, and Robert wanted to find out, so he spent months doing a full implementation of Brendan McKay’s algorithm (the only other one). This had absolutely nothing to do with his Ph.D. thesis work on the Birch and Swinnerton-Dyer conjecture, but I was very supportive.

Craig Citro (Google)

Craig Citro did a Ph.D. in number theory (with Hida), but also worked on Sage aLOT as a grad student and postdoc. He’s done a lot of hiring at Google. He says: “My main piece of advice to potential google applicants is ‘start writing as much code as you can, right now.’ Find out whether you’d actually enjoyworking for a company like Google, where a large chunk of your job may be coding in front of a screen. I’ve had several friends from math discover (the hard way) that they don’t really enjoy full-time programming (any more than they enjoy full-time teaching?).”
“Start throwing things on github now. Potential interviewers are going to check out your github profile; having some cool stuff at the top is great, but seeing a regular stream of commits is also a useful signal.”

Robert Bradshaw (Google)

“A lot of mathematicians are good at (and enjoy) programming. Many of them aren’t (and don’t). Find out. Being involved in Sage is significantly more than just having taken a suite of programming courses or hacking personal scripts on your own: code reviews, managing bugs, testing, large-scale design, working with others’ code, seeing projects through to completion, and collaborating with others, local and remote, on large, technical projects are all important. It demonstrates your passion.”

Rado Kirov (Google)

Robert Bradshaw said it before me, but I have to repeat. Large scale software development requires exposure to a lot of tooling and process beyond just writing code - version control, code reviews, bug tracking, code maintenance, release process, coordinating with collaborators. Contributing to an active open-source project with a large number of contributors like Sage, is a great way to experience all that and see if you would like to make it your profession. A lot of mathematicians write clever code for their research, but if less than 10 people see it and use it, it is not a realistic representation of what working as a software engineer feels like. 

The software industry is in large demand of developers and hiring straight from academia is very common. Before I got hired by Google, the only software development experience on my resume was the Sage graph editor. Along with solid understanding of algorithms and data structures that was enough to get in."

David Moulton (Google)

“Google hires mathematicians now as quantitative analysts = data engineers. Google is very flexible for a tech company about the backgrounds of its employees. We have a long-standing reading group on category theory, and we’re about to start one on Babai’s recent quasi- polynomial-time algorithm for graph isomorphism. And we have a math discussion group with lots of interesting math on it.”

My advice for math professors

Obviously, encourage your students to get involved in open source projects like Sage, even if it appears to be a waste of time or distraction from their thesis work (this will likely feel very counterintuitive you’ll hate it).
At Univ of Washington, a few years ago I taught a graduate-level course on Sage development. The department then refused to run it again as a grad course, which was frankly very frustrating to me. This is exactly the wrong thing to do if you want to increase the options of your Ph.D. students for industry jobs. Maybe quit trying to train our students to be only math professors, and instead give them a much wider range of options.

by William Stein ([email protected]) at January 08, 2016 10:25 PM