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Just so I have this excellent article link to hand: ASP.NET 2.0 Security Practices at a Glance

Keith Devlin and Jonathan Borwein. AK Peters, 2008, ISBN-13: 978-1568813431

Whenever a book’s preface states its aims, a natural question to ask is whether it succeeds in meeting them. Keith Devlin and Jonathan Borwein, two mathematicians with expertise in different mathematical fields but with a common interest in experimental mathematics, begin this book by saying:

“Our aim in writing this book was to provide a short, readable account of experimental mathematics. It is not intended as a textbook to accompany a course…In particular, we do not aim for comprehensive coverage of the field; rather, we pick and choose topics and examples to give the reader a good sense of the current state of play in the rapidly growing field of experimental mathematics”

The sleuth-like style and lucid writing certainly make this book an enjoyable read. Many explanations are framed by relevant historical context and tales of mathematicians whose use of experimental mathematics helped them gain insights into difficult problems. Although it was never intended to be a course textbook, it could be used as a supplementary text. Many of the chapters are short, and should be viewed as aperitifs.

Chapter 1 deals with the important question “What is Experimental Mathematics?”. In the authors’ own words,

“Experimental mathematics is the use of a computer to run computations – sometimes no more than trial-and-error tests – to look for patterns, to identify particular numbers and sequences, to gather evidence in support of specific mathematical assertions that may themselves arise by computational means, including search”.

Broadly speaking, it is the use of computers in mathematics as tools in their own right, not simply as numerical calculation aids, “…experimentation is regarded as a significant part of mathematics in its own right…”.

What kind of experimentation? Here are some of the things described in this book:

• Symbolic computation using a computer algebra system such as Maple or Mathematica
• Data Visualisation
• Integer-relation algorithms like PSLQ
• High precision integer and floating point arithmetic
• High precision evaluation of integrals and summation of infinite series
• Identification of functions based on their graph characteristics

It would be very easy to fall into the belief that great mathematicians pluck profound and deep results out of thin air, but some of the mathematical greats (Gauss, Euler, Fermet, Riemann…) were confirmed experimenters who would spend many hours carrying out calculations in order to discover new mathematical avenues worth pursuing. The 72 year old Gauss recounted in a letter to the astronomer, Johann Encke, that as a young boy of 15, armed with a table of logarithms he ‘frequently spent an idle quarter of an hour to count another chiliad here and there’ [3], which led to his estimate of the density of prime numbers; “..Gauss was very clearly an ‘experimental mathematician’ of the first order.”

Chapter 2 gives a brief introduction to the PSLQ algorithm, an integer relation algorithm developed by Helaman Ferguson. Given any real coefficients a₀, a₁ ,…, a_n and a precision ε, an integer relation algorithm uses high-precision arithmetic to find integer coefficients λ₀, λ₁, λ₂, …, λ_n such that λ₀ ≠ 0 and

|λ₀a₀ + λ₁a₁ + … + λ_na_n| < ε

or else it tells you such expression exists within a ball of a given radius about the origin.

Chapter 3 (What Is That Number?) introduces Inverse Symbolic Calculators as tools to recognise numbers, and combined with Sloane’s online Encyclopedia of Integer Sequences, describes a technique for determining closed forms of sequences.

I have more than a passing interest in Riemann’s zeta function, the topic of Chapter 4 (The Most Important Function in Mathematics); I found it interesting though perhaps a little short. I particularly liked the quote about British soccer player, Wayne Rooney, contrasting him with David Beckham: “There is more chance of him [Rooney] proving Riemann’s Hypothesis than wearing a sarong”!

I’m certain physicists will find chapter 5 (Evaluate the Following Integral) interesting, especially given the authors’ collaborations in computing closed forms of definite integrals arising in physics.

Chapter 9 (Take It to the Limit) contains 3 worked examples of finding closed forms for infinite sums, and Chapter 10 (Danger! Always Exercise Caution When Using the Computer) contains sobering stories and examples of some of the pitfalls faced by experimental mathematicians. Here is one:

A computer algebra system (CAS) will discover that I₁ = I₂ = I₃ = … = I₇ = π/2 but I₈ = 0.499999999992646π

On finding this, the authors suspected a bug in the CAS software. But there is no bug!

The book provides tantalising examples and suggestions to whet the reader’s appetite in the form of an ‘Explorations’ section at the end of each chapter, not exactly exercises but there is a corresponding ‘Answers and Reflections’ chapter at the end of the book. Interested readers will find many of these topics expanded upon in [1].

I thoroughly enjoyed reading this short introduction to experimental mathematics. It will no doubt appeal to a broad mathematical audience, both professional and amateur alike. If I have one complaint, well more of a request, it would be a much longer chapter on evaluating definite integrals! But then, in the words of G. H. Hardy, “I could never resist a definite integral” [2].

 

References

[1] David Bailey et al. (2007). Experimental Mathematics in Action. AK Peters, MA.

[2] Macdonald, H. S. (1999). The Beauty of Geometry: Twelve Essays. Dover Publications.

[3] Havil, J. (2003). Gamma: Exploring Euler’s Constant. Princeton UP.

[This review appeared in the July 2009 issue of the Australian Mathematical Society’s Gazette.]

Did you know that SQL Server 2008 has the ability to retrieve deadlock information after the fact without having previously enabled any additional tracing?  It’s part of the new advanced troubleshooting feature called Extended Events. Here’s a SQL query from Jonathan Kehayias which retrieves SQL Server 2008 deadlock information:

select CAST( REPLACE( REPLACE(XEventData.XEvent.value(‘(data/value)[1]’, ‘varchar(max)’), ”, ”), ”,”) as xml) as DeadlockGraph FROM (select CAST(target_data as xml) as TargetData from sys.dm_xe_session_targets st join sys.dm_xe_sessions s on s.address = st.event_session_address where name = ‘system_health’) AS Data CROSS APPLY TargetData.nodes (‘//RingBufferTarget/event’) AS XEventData (XEvent) where XEventData.XEvent.value(‘@name’, ‘varchar(4000)’) = ‘xml_deadlock_report’

More info. here: Using SQL Server 2008 Extended Events

I blogged about ClearTrace a while back. There’s a new version available that targets SQL Server 2008 (as well as SQL Server 2005). Download here. It’s a useful tool and the improved performance loading large trace files is a bonus.

Just so I remember where this useful page of SQL Server maximum values is: Maximum Capacity Specifications for SQL Server.

Join us at the Perth .NET User Group, Thurs July 2nd to hear Jeremy Thake answer a common question: Is SharePoint is the correct choice as a base platform for a solution? Jeremy Thake will present the advantages (and disadvantages) of leveraging the SharePoint Platform. Some lines in the sand will be drawn to make it easier to make the decision early on the project based on real world experience gathered by Jeremy from the SharePoint community.

TOPIC: When should ASP.NET Developers leverage the SharePoint Platform?
DATE: Thursday, July 2nd, 5:30pm
VENUE: Excom, Ground Floor, 23 Barrack Street, Perth
COST: Free. All welcome

More details here.

Ola Hallengren has updated his SQL Server maintenance scripts with several new features:

• Rebuild and reorganize partitioned indexes on the partition level in IndexOptimize. Just set the new parameter @PartitionLevel = ‘Y’.
• Exclusion of indexes on read-only filegroups in IndexOptimize.
• Use LiteSpeed compression in DatabaseBackup. Just set the new parameter @BackupSoftware = ‘LITESPEED’.
• New feature to run the stored procedures without executing the commands. Just use the parameter @Execute = ‘N’.
• The solution now works on all builds and editions of SQL Server 2005 and SQL Server 2008

Well worth checking out.

Ever needed a HTML parser that is very tolerant of “real world” malformed HTML, builds a read/writable DOM and supports plain XPATH or XSLT? Sounds like the HTML Agility Pack .NET library. The object model is very similar to XmlDocument.

Among the myriad of possible applications, it can be used for page fixups, generating pages, web scrapers, etc.

Brad McGehee has posted a nice introduction to SQL Server 2008 indexes over at Simple-Talk: Brad’s Sure Guide to Indexes including a description of the new Filtered Index in SQL Server 2008.

If you want a quick and easy way to check SQL Server connectivity, without resorting to the full blown functionality of WireShark (or similiar), you can use Microsoft’s PortQryUI utility to troubleshoot TCP/IP connectivity problems. PortQryUI is a GUI on top of the PortQry command line tool, with predefined groups of ports to scan. One of the these groups targets SQL Server, which consists of UDP port 1434 and TCP port 1433. To check these ports, enter the IP address (or fully qualified domain name) of the target SQL Server instance, select SQL Service in the ‘Service to Query’ drop-down box and click the Query button.

If the ports are OK, the utility will list the ports as Listening. Otherwise, it will tell you the ports are being Filtered or are Not Listening.

You can download PortQryUI from here: PortQryUI – User Interface for the PortQry Command Line Port Scanner.