<p>Anyone care to divulge on the comparisons/contrasts of discrete, linear, and abstract mathematics? I am already vaguely familiar, but - would appreciate the opportunity to receive feedback from students/professionals and the like, of whom have had hands-on experience. Also, feel free to throw in any other related math disciplines of which may contribute to the relevance of the discussion.</p>
<p>discrete (usually vs continous) involves things like counting, combinitorics, number theory, cryptography, etc. Basically, things that are not continuous. Calculus is not a subject in discrete mathematics. Algebra is. </p>
<p>linear involves any kind of operation T() where T(ax+by) = aT(x)+bT(y). T can be a matrix, a function, or some other kind of operation. These problems are really important because nonlinear problems are often approximated by linear ones because linear problems often have tractable closed form solutions, and nonlinear problems often don’t. </p>
<p>Abstract (usually vs Applied). Abstract mathematics is basically about proving theorems. Applied is usually about solving problems. </p>
<p>These are not disciplines per se, but are properties certain areas of mathematics possess. You can work in an area that is simultaneously discrete, linear, and abstract.</p>
<p>ClassicRockerDad, can you cite a source for this:</p>
<p>T(ax+by) = aT(x)+bT(y)</p>
<p>being the definition of a linear function. It seems like you are also saying
T(ax+by)=xT(a)+yT(b) assuming [a,x]=[b,y]=0</p>
<p>This seems like a more specific, and weird, definition of a linear function than I am used to.</p>
<p>[Linear</a> Function – from Wolfram MathWorld](<a href=“http://mathworld.wolfram.com/LinearFunction.html]Linear”>Linear Function -- from Wolfram MathWorld)
[Linear</a> function - Wikipedia, the free encyclopedia](<a href=“http://en.wikipedia.org/wiki/Linear_function]Linear”>Linear function - Wikipedia)</p>
<p>What I wrote follows from the definitions in ucbalumnus’s citations. I’ll use f instead of T. </p>
<p>f(ax+by)=f(ax)+f(by)=af(x)+bf(y).</p>