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I am playing around, writing a program that will rotate a set of points in 3d space and I am using a bit of trigonometry for this. When I think about a 90 degree rotation visually, it seems natural that cos(pi/2) * tan(pi/2) should equal 1, and my program depends on that result. Luckily, that’s the answer that python computes. Even in an Excel spreadsheet, that’s the answer. However, analytically, the answer should be 0, since cos(pi/2) is 0. Or at least the answer should be undefined, since tan(pi/2) is infinity.
So I guess that’s the mathematical answer, that the value is undefined. But my question is, is it conventional to just agree that the answer is 1? Is that why computer programs will return a value of 1? Or is the value of 1 returned only because of technical reasons, in that cos(pi/2) is actually stored as some minuscule non-zero number and tan(pi/2) is stored as some large but finite number? Or is it both, in that the computer would return a value of 1 for unintended technical reasons, but we all agree that it’s ok, because we agree to define cos(pi/2) * tan(pi/2) as 1?
I recently got into complex analysis and I often see people use the residue theorem to solve real integrals from -∞ to ∞.
One technique that I often see people use is, let gamma 1 be the parametrization of a semicircle with radius R on the upper-half complex plane, and let gamma 2 be the line from R to -R. Combining these two curves gives us a intagral over a closed curve, which we can use the residue theorem on.
They then use the estimation theorem to prove that the integral over gamma 1 is zero, which means that the integral over gamma is equal to the original integral. My question is, what if gamma 1 is nonzero? Is there some other techniques to evaluate integrals of such kind?
On Monday I have to present an exam which will mainly be about showing and proving results, particularly showing if a set is convex, if a function is (quasi)concave/convex, proving by direct argument if a point is a miximum. We can use the textbook and our own notes so memorizing theorems won’t be a problem, so do you have any suggestions?
Hi, I am doing a binary logistic regression on smoking data from a well known study. I am doing the BLR with 25 year survival status as the DV (Alive=0, Dead=1) and smoking status (No=0, Yes=1) When conducting the BLR I am asked to first do the smoking data only and then the second analysis is to include age as a variable (categorized into 5 groups). The first run with only the smoking variable I get a Exp(B) of 1.46, which I believe means that a smoker is 1.46 times more likely to be in the Dead group. No running the analysis with the interaction of smoking status and age (both categorical) I get -B’s in all categories and (Exp) B ranging from .025, .051,.095,.282,.785,.645. Am I correct in saying that when age is considered that the likelihood of smokers belonging in the death group is actually reduced? I am using an interaction, should I not use an interaction between the covariables?? Thanks!
My only motivation for learning Abstract Algebra is to have examples to play around with for category theory.
I am thinking of self-studying the first two chapters of Aluffi’s Algebra: Chapter 0, which covers basic set theory, category theory (basics) and group theory, up to Lagrange’s theorem. I’m already familiar with abstract linear algebra from Axler.
So would linear algebra and group theory be sufficient to learn about and appreciate category theory? Or should I learn more abstract algebra? How much topology should I know?
My interest in learning category theory is in foundations of mathematics and type theory.