Archive for the ‘differential equations’ Category

Book presentation at the USC Python Users Group

December 7, 2013 Leave a comment

Areas of Mathematics

October 22, 2013 3 comments

For one of my upcoming talks I am trying to include an exhaustive mindmap showing the different areas of Mathematics, and somehow, how they relate to each other. Most of the information I am using has been processed from years of exposure in the field, and a bit of help from Wikipedia.

But I am not entirely happy with what I see: my lack of training in the area of Combinatorics results in a rather dry treatment of that part of the mindmap, for example. I am afraid that the same could be told about other parts of the diagram. Any help from the reader to clarify and polish this information will be very much appreciated.

And as a bonus, I included a \LaTeX script to generate the diagram with the aid of the tikz libraries.

\tikzstyle{level 2 concept}+=[sibling angle=40]
\begin{tikzpicture}[scale=0.49, transform shape]
  \path[mindmap,concept color=black,text=white]
    node[concept] {Pure Mathematics} [clockwise from=45]
      child[concept color=DeepSkyBlue4]{
        node[concept] {Analysis} [clockwise from=180]
          child { 
            node[concept] {Multivariate \& Vector Calculus}
              [clockwise from=120]
              child {node[concept] {ODEs}}}
              child { node[concept] {Functional Analysis}}
              child { node[concept] {Measure Theory}}
              child { node[concept] {Calculus of Variations}}
              child { node[concept] {Harmonic Analysis}}
              child { node[concept] {Complex Analysis}}
              child { node[concept] {Stochastic Analysis}}
              child { node[concept] {Geometric Analysis}
                [clockwise from=-40]
                child {node[concept] {PDEs}}}}
          child[concept color=black!50!green, grow=-40]{ 
            node[concept] {Combinatorics} [clockwise from=10]
              child {node[concept] {Enumerative}}
              child {node[concept] {Extremal}}
              child {node[concept] {Graph Theory}}}
          child[concept color=black!25!red, grow=-90]{ 
            node[concept] {Geometry} [clockwise from=-30]
              child {node[concept] {Convex Geometry}}
              child {node[concept] {Differential Geometry}}
              child {node[concept] {Manifolds}}
              child {node[concept,color=black!50!green!50!red,text=white] {Discrete Geometry}}
              child {
                node[concept] {Topology} [clockwise from=-150]
                  child {node [concept,color=black!25!red!50!brown,text=white]
                    {Algebraic Topology}}}}
          child[concept color=brown,grow=140]{ 
            node[concept] {Algebra} [counterclockwise from=70]
              child {node[concept] {Elementary}}
              child {node[concept] {Number Theory}}
              child {node[concept] {Abstract} [clockwise from=180]
                child {node[concept,color=red!25!brown,text=white] {Algebraic Geometry}}}
              child {node[concept] {Linear}}}
    node[extra concept,concept color=black] at (200:5) {Applied Mathematics} 
      child[grow=145,concept color=black!50!yellow] {
        node[concept] {Probability} [clockwise from=180]
          child {node[concept] {Stochastic Processes}}}
      child[grow=175,concept color=black!50!yellow] {node[concept] {Statistics}}
      child[grow=205,concept color=black!50!yellow] {node[concept] {Numerical Analysis}}
      child[grow=235,concept color=black!50!yellow] {node[concept] {Symbolic Computation}};

Have a child, plant a tree, write a book

April 23, 2013 2 comments

Or more importantly: rear your children to become nice people, water those trees, and make sure that your books make a good impact.

I recently enjoyed the rare pleasure of having a child (my first!) and publishing a book almost at the same time. Since this post belongs in my professional blog, I will exclusively comment on the latter: Learning SciPy for Numerical and Scientific Computing, published by Packt in a series of technical books focusing on Open Source software.

Keep in mind that the book is for a very specialized audience: not only do you need a basic knowledge of Python, but also a somewhat advanced command of mathematics/physics, and an interest in engineering or scientific applications. This is an excerpt of the detailed description of the monograph, as it reads in the publisher’s page:

It is essential to incorporate workflow data and code from various sources in order to create fast and effective algorithms to solve complex problems in science and engineering. Data is coming at us faster, dirtier, and at an ever increasing rate. There is no need to employ difficult-to-maintain code, or expensive mathematical engines to solve your numerical computations anymore. SciPy guarantees fast, accurate, and easy-to-code solutions to your numerical and scientific computing applications.

Learning SciPy for Numerical and Scientific Computing unveils secrets to some of the most critical mathematical and scientific computing problems and will play an instrumental role in supporting your research. The book will teach you how to quickly and efficiently use different modules and routines from the SciPy library to cover the vast scope of numerical mathematics with its simplistic practical approach that is easy to follow.

The book starts with a brief description of the SciPy libraries, showing practical demonstrations for acquiring and installing them on your system. This is followed by the second chapter which is a fun and fast-paced primer to array creation, manipulation, and problem-solving based on these techniques.

The rest of the chapters describe the use of all different modules and routines from the SciPy libraries, through the scope of different branches of numerical mathematics. Each big field is represented: numerical analysis, linear algebra, statistics, signal processing, and computational geometry. And for each of these fields all possibilities are illustrated with clear syntax, and plenty of examples. The book then presents combinations of all these techniques to the solution of research problems in real-life scenarios for different sciences or engineering — from image compression, biological classification of species, control theory, design of wings, to structural analysis of oxides.

The book is also being sold online in Amazon, where it has been received with pretty good reviews. I have found other random reviews elsewhere, with similar welcoming comments:

What if?

April 11, 2012 Leave a comment

Sara, Spencer and Cam (left to right) discussing their projects in office hour

What if?” is a truly powerful question. It is the question that separates the child from the adult; the student from the professional. Average students will go through the motions of a course and ask themselves many times: “What is the point of all this?” On the other hand, the notable students on their way to excellence will ask themselves: “How can I profit from this?” It is these inquiring minds who make it at the end: they are a pleasure to work with, they have the drive and the passion to get the job done, enjoy the process, and they are more likely to give their future employers more of their time in pursue of solutions—not because there is monetary or status gain alone, but because their commitment is only matched with their skill and curiosity.

It is a thrill to witness your own students pose that “What if?” question to themselves, and take steps to accomplish that little dream with the knowledge obtained in class. This semester, I had been blessed with a group of extremely talented people in all my different teaching assignments—especially those in my course on elementary differential equations.

Read more…

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