I have a very hard time with math yet I always seem to find myself in ever increasingly difficult math courses. Over the years I have amassed a small 'fortune' of various scripts and programs. They are written in a variety of languages, from C to Java to Python and even a few in Lisp (I like to fool around with different languages). These are great for getting fast answers or checking my homework, but not so great for showing how those answers were gotten. My question is how would I go about adding a 'solver' aspect to these various programs? It doesn't need to be language specific, but just a concept or maybe a specific word or definition would be of immense help. I basically want to do a Wolfram Alpha type program, where you punch in the problem and it shows a step by step solution. I have spent the last couple of days researching this, but I am always redirected towards an app or, frustratingly, to Wolfram Alpha itself. Any suggestions would be much appreciated. Thank you.
What categories of math are you talking about?
There does not seem to be a universal math solving program, what you mentioned sounds to me like the concept of a complicated math system(like MatLab).
And for, I think, a large scale of mathematical problems, the algorithms for computer programs to solve them differ from our ways to solve them.
Rather than a universal algorithm to achieve your goal, I think you can divide your problem into pieces, and achieve them with different sets of programs, there is a thesis Symbolic Integration by Joel Moses, the first chapters give a good view of how to divide a math problem into programmable cases(in the example of solving quadratic equations), and the last hundred pages is a set of algorithms for solving irrational integration, interpreted in LISP.
I just want to get the main idea/principle of openFOAM and how you create a simulation, please let me know where I go wrong,
So basically you have a object that interacts with gas or liquid and you want to simulate this, so you create model of the object, mesh it, specify where the gas will flow in and out and what are the walls, and set the other correct parameters and then run the program (with the approprate time step etc)?
OpenFOAM is an open source C++ library which implements the finite volume method (FVM), which is widely used in CFD.
What you have explained is a vague understanding of some of the applications of CFD. Those things you specified might not always be the case (i.e. the fluid might not necessarily be (a) gas and so on.
The main stages of a CFD problem are: making the geometry - mesh generation - preprocess - solving - postprocess.
There might be more stages added depending on the resolution and other specifics of the case.
Now OpenFoam is an open source (free for all) tool which is in C++ and helps solve the CFD problems. If the problem is simple and routine, and you have access to a commercial solver such as ANSYS fluent, then you can use that since it is easier and much less work if the problem is not specific. However, if the problem is specific and there are customized criteria OpenFoam is a nice tool.
It is written in C++ thus it is object oriented and also there are many many different solvers already written and available to use, so you will not have to write all the schemes and everything on your own from scratch.
However, my main advice to you is to read more about CFD to have a clear understanding, there are tens of good books avaiable.
I'm giving TDD a serious try today and have found it really helpful, in line with all the praise it receives.
In my quest for exercises on which to learn both Python and TDD i have started to code SPOJ exercises using the TDD technique and i have arrived at a question:
Given that all of SPOJ's exercises are mostly math applied to programming; How does one test a math procedure as in the TDD Fashion? Sample known-to-be-correct data? Test against a known implementation?
I have found that using the sample data given in the problem itself is valuable but it feels overkill for something you can test so quickly using the console, not to mention the overhead to design your algorithms in a testable fashion (Proxying the stdout and stdin objects is nothing short of too much work for a really small reward), and while it is good because it forces you to think your solutions in testable terms i think i might be trying way too hard on this.
All guidance is welcome
test all edge cases. Your program is most likely to fail when the input is special for some reason: negative, or zero values, very large values, inputs in reverse order, empty inputs. You might also want some destructive tests to see how large the inputs can be before things break or grind to a halt.
Sphere online Judge may not be the best fit of TDD. For one the input data might be better behaved than what a real person might put in. Secondly there is a code size limit on some problems. An extensive test-suite might put you over that limit.
You might want to take a look at Uncle Bob's "Transformation Priority Premise." It offers some guidance on how to pick a sequence of tests to test drive an algorithm.
Use sample inputs for which you know the results (outputs). Use equivalence partitioning to identify a suitable set of test cases. With maths code you might find that you can not implement as incrementally as for other code: you might need several test cases for each incremental improvement. By that I mean that non maths code can typically be thought of as having a set of "features" and you can implement one feature at a time, but maths code is not much like that.
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I haven't taken any math classes above basic college calculus. However, in the course of my programming work, I've picked up a lot of math and comp sci from blogs and reading, and I genuinely believe I have a decent mathematical mind. I enjoy and have success doing Project Euler, for example.
I want to dive in and really start learning some cool math, particularly discrete mathematics, set theory, graph theory, number theory, combinatorics, category theory, lambda calculus, etc.
My impression so far is that I'm well equipped to take these on at a conceptual level, but I'm having a really hard time with the mathematical language and symbols. I just don't "speak the language" and though I'm trying to learn it, I'm the going is extremely slow. It can take me hours to work through even one formula or terminology heavy paragraph. And yeah, I can look up terms and definitions, but it's a terribly onerous process that very much obscures the theoretical simplicity of what I'm trying to learn.
I'm really afraid I'm going to have to back up to where I left off, get a mid-level math textbook, and invest some serious time in exercises to train myself in that way of thought. This sounds amazingly boring, though, so I wondered if anyone else has any ideas or experience with this.
If you don't want to attend a class, you still need to get what the class would have given you: time in the material and lots of practice.
So, grab that text book and start doing the practice problems. There really isn't any other way (unless you've figured out how osmosis can actually happen...).
There is no knowledge that can only be gained in a classroom.
Check out the MIT Courseware for Mathematics
Also their YouTube site
Project Euler is also a great way to think about math as it relates to programming
Take a class at your local community college. If you're like me you'd need the structure. There's something to be said for the pressure of being graded. I mean there's so much to learn that going solo is really impractical if you want to have more than just a passing nod-your-head-mm-hmm sort of understanding.
Sounds like you're in the same position I am. What I'm finding out about math education is that most of it is taught incorrectly. Whether a cause or result of this, I also find most math texts are written incorrectly. Exceptions are rare, but notable. For instance, anything written by Donald Knuth is a step in the right direction.
Here are a couple of articles that state the problem quite clearly:
A Gentle Introduction To Learning
Calculus
Developing Your Intuition For
Math
And here's an article on a simple study technique that aims at retaining knowledge:
Teaching linear algebra
Consider auditing classes in discrete mathematics and proofs at a local university. The discrete math class will teach you some really useful stuff (graph theory, combinatorics, etc.), and the proofs class will teach you more about the mathematical style of thinking and writing.
I'd agree with #John Kugelman, classes are the way to go to get it done properly but I'd add that if you don't want to take classes, the internet has many resources to help you, including recorded lectures which I find can be more approachable than books and papers.
I'd recommend checking out MIT Open Courseware. There's a Maths for Computer Science module there, and I'm enjoying working through Gilbert Strang's Linear Algebra course of video lectures.
Youtube and videolectures.com are also good resources for video lectures.
Finally, there's a free Maths for CS book at bookboon.
To this list I would now add The Haskel Road to Logic, Maths, and Programming, and Conceptual Mathematics: A First Introduction to Categories.
--- Nov 16 '09 answer for posterity--
Two books. Diestel's Graph Theory, and Knuth's Concrete Mathematics. Once you get the hang of those try CAGES.
Find a good mentor who is an expert in the field who is willing to spend time with you on a regular basis.
There is a sort of trick to learning dense material, like math and mathematical CS. Learning unfamiliar abstract stuff is hard, and the most effective way to do it is to familiarize yourself with it in stages. First, you need to skim it: don't worry if you don't understand everything in the first pass. Then take a break; after you have rested, go through it again in more depth. Lather, rinse, repeat; meditate, and eventually you may become enlightened.
I'm not sure exactly where I'd start, to become familiar with the language of mathematics; I just ended up reading through lots of papers until I got better at it. You might look for introductory textbooks on formal mathematical logic, since a lot of math (especially in language theory) is based off of that; if you learn to hack the formal stuff a bit, the everyday notation might look a bit easier.
You should probably look through books on topics you're personally interested in; the inherent interest should help get you over the hump. Also, make sure you find texts that are actually introductory; I have become wary of slim, undecorated hardbacks labeled Elementary Foobar Theory, which tend to be elementary only to postdocs with a PhD in Foobar.
A word of warning: do not start out with category theory -- it is the most boring math I have ever encountered! Due to its relevance to language design and type theory, I would like to know more about it, but so far I have not been able to deal...
For a nice, scattershot intro to bits of many kinds of CS-ish math, I recommend Godel, Escher, Bach by Hofstadter (if you haven't read it already, of course). It's not a formal math book, though, so it won't help you with the familiarity problem, but it is quite inspirational.
Mathematical notation is is akin to several computer languages:
concise
exacting
based on many idioms
a fair amount of local variations and conventions
As with a computer language, you don't need to "wash the whole elephant at once": take it one part a at time.
A tentative plan for you could be
identify areas of mathematics that are interesting or important to you. (seems you already have a bit of a sense for that, CS has helped you develop quite a culture for it.)
take (or merely audit) a few formal classes in this area. I agree with several answers in this post, an in-person course, at local college is preferable, but, maybe at first, or to be sure to get the most of a particular class, first self-teaching yourself in this area with MIT OCW, similar online resources and associated books is ok/fine.
if an area of math introduces too high of a pre-requisite in terms of fluency with notation or with some underlying concept or (most often mechanical computation and transformation techniques). No problem! Just backtrack a bit, learn these foundations (and just these foundations!) and move forward again.
Find a "guru", someone that has a broad mathematical culture and exposure, not necessarily a mathematician, physics folks are good too, indeed they can often articulate math in a more practical fashion. Use this guru to guide you, as he/she can show you how the big pieces fit together.
Note: There is little gain to be had of learning mathematical notation for its own sake. Rather it should be learned in context, just like say a C# idiom is better memorized when used and when associated with a specific task, rather than learned in vacuo. A related SO posting however provides several resources to decipher and learn mathematical notation
Project Euler takes problems out of context and drops them in for people to solve them. Project Euler cannot teach you anything effectively. I think you should forget about it, if it is popular it does not mean anything. You cannot study Mathematics through Project Euler as it contains only bits and pieces(and some pretty high level pieces) that you're supposed to know in order to solve the problems. Learning mathematics means to consider a subject and a read a book about it and solving exercices or reading solutions, that's how you learn math. If it so happens that through your reading you find something that is close to some project euler thing, your luck , but otherwise Project euler is a complete waste of time. I think the time is much better invested choosing a particular branch of mathematics and studying that. Let me explain why: I solved 3 pretty advanced Projec Euler problems and they were all making appeal to knowledge from Number theory which I happened to have because i studies some part of it. I do not think Iearned anything from Project Euler, it just happened that I already knew some number theory and solved the problems.
For example, if you find out you like number theory, take H. Davenport -> Hardy & Wright -> Kenneth & Rosen's , study those.
If you like Graph Theory take Reinhard Diestel's book which is freely available and study that(or check books.google.com and find whichever is more appropriate to your taste) but don't spread your attention in 999999 directions just because Project Euler has problems ranging from dynamic programming to advanced geometry or to advanced number theory, that is clearly the wrong way to go and it will not bring you closer to your goal.
This sounds amazingly boring
Well ... Mathematics is not boring when you find some problem that you are attached to, which you like and you'd like to find the solution to, and when you have the sufficient time to reflect on it while not behind a computer screen. Mathematics is done with pen and paper mostly(yes you can use computers .. but that's not really the point).
So, if you find a real-world problem, or some programming problem that would benefit from
you knowing some advanced maths, and you know what maths you have to study , it can be motivating to learn in that way.
If you feel you are not motivated it is hard to study properly.
There is also the question of what you actually mean when you say learn. Does the learning process stop after you solved the problems at the end of the chapter of a book ? Well you decide. You can consider you have finished learning that subject, or you can consider you have not finished and read more about it. There are entire books on just one equation and variations of it.
The amount of programming-related math that you can learn without formal training is limited, but it's more than enough. But maybe you can self-teach yourself.
It all boils down to your resources and motivation.
To know mathematics you have to do mathematics not programming(project euler).
For beginning to learn category theory I recommend David Spivak's Category Theory for the Sciences (AKA Category Theory for Scientists) because its relatively comprehensible due to many examples that enable understanding by analogy and which quickly builds a foundation for understanding more abstract concepts.
It requires the ability to reason logically and an intuitive notion of what is a set. It proceeds from sets and functions through basic category theory to adjoint functors, categories of functors, sheaves, monads and an introduction to operads. Two main threads throughout are modeling databases in terms of categories and describing categories with annotated diagrams called ologs. The bibliography provides references to more advanced and specialized topics including recent papers by Dr. Spivak.
An expected outcome from reading this book is the capability of understanding category theory texts and papers written for mathematicians such as Mac Lane's Category Theory for the Working Mathematician.
In PDF format it is available from http://math.mit.edu/~dspivak/teaching/sp13/ (the dynamic version is recommended since its the most recent). The open access HTML version is available from https://mitpress.mit.edu/books/category-theory-sciences (which is recommended since it includes additional content including answers to some exercises).
I'm interested in learning more about pattern recognition. I know that's somewhat of a broad field, so I'll list some specific types of problems I would like to learn to deal with:
Finding patterns in a seemingly random set of bytes.
Recognizing known shapes (such as circles and squares) in images.
Noticing movement patterns given a stream of positions (Vector3)
This is a new area of experimentation for me personally, and to be honest, I simply don't know where to start :-) I'm obviously not looking for the answers to be provided to me on a silver platter, but some search terms and/or online resources where I can start to acquaint myself with the concepts of the above problem domains would be awesome.
Thanks!
ps: For extra credit, if said resources provide code examples/discussion in C# would be grand :-) but doesn't need to be
Hidden Markov Models are a great place to look, as well as Artificial Neural Networks.
Edit: You could take a look at NeuronDotNet, it's open source and you could poke around the code.
Edit 2: You can also take a look at ITK, it's also open source and implements a lot of these types of algorithms.
Edit 3: Here's a pretty good intro to neural nets. It covers a lot of the basics and includes source code (albeit in C++). He implemented an unsupervised learning algorithm, I think you may be looking for a supervised backpropagation algorithm to train your network.
Edit 4: Another good intro, avoids really heavy math, but provides references to a lot of that detail at the bottom, if you want to dig into it. Includes pseudo-code, good diagrams, and a lengthy description of backpropagation.
This is kind of like saying "I'd like to learn more about electronics.. anyone tell me where to start?" Pattern Recognition is a whole field - there are hundreds, if not thousands of books out there, and any university has at least several (probably 10 or more) courses at the grad level on this. There are numerous journals dedicated to this as well, that have been publishing for decades ... conferences ..
You might start with the wikipedia.
http://en.wikipedia.org/wiki/Pattern_recognition
This is kind of an old question, but it's relevant so I figured I'd post it here :-) Stanford began offering an online Machine Learning class here - http://www.ml-class.org
OpenCV has some functions for pattern recognition in images.
You might want to look at this :http://opencv.willowgarage.com/documentation/pattern_recognition.html. (broken link: closest thing in the new doc is http://opencv.willowgarage.com/documentation/cpp/ml__machine_learning.html, although it is no longer what I'd call helpful documentation for a beginner - see other answers)
However, I also recommend starting with Matlab because openCV is not intuitive to use.
Lot of useful links on this page on computer vision related pattern recognition. Some of the links seem to be broken now but you may find it useful.
I am not an expert on this, but reading about Hidden Markov Models is a good way to start.
Beware false patterns! For any decently large data set you will find subsets that appear to have pattern, even if it is a data set of coin flips. No good process for pattern recognition should be without statistical techniques to assess confidence that the detected patterns are real. When possible, run your algorithms on random data to see what patterns they detect. These experiments will give you a baseline for the strength of a pattern that can be found in random (a.k.a "null") data. This kind of technique can help you assess the "false discovery rate" for your findings.
learning pattern-recoginition is easier in matlab..
there are several examples and there are functions to use.
it is good for the understanding concepts and experiments...
I would recommend starting with some MATLAB toolbox. MATLAB is an especially convenient place to start playing around with stuff like this due to its interactive console. A nice toolbox I personally used and really liked is PRTools (http://prtools.org); they have an implementation of pretty much every pattern recognition tool and also some other machine learning tools (Neural Networks, etc.). But the nice thing about MATLAB is that there are many other toolboxes as well you can try out (there is even a proprietary toolbox from Mathworks)
Whenever you feel comfortable enough with the different tools (and found out which classifier is perfomring best for you problem), you can start thinking about implementing the machine learning in a different application.