顯示具有 stage 3 標籤的文章。 顯示所有文章
顯示具有 stage 3 標籤的文章。 顯示所有文章

Stage 3 Probability and Stochastic (Random) Processes

Probability and Stochastic (Random) Processes (optional):
Conditional expectation, Poisson process, Markov chains, renewal theory, queueing theory, reliability theory, Brownian motion and stationary processes. This topic is particularly useful for electrical and computer engineers, actuarial study, finance or things like that. See also left material from stage 2 probability. If you want to learn this with measure, see stage 4 probability list.
沒有留言:
標籤：

Stage 3 Statistical Computing

Statistical Computing (optional):
These days, you can't do statistics without computer, in practice. R is a freeware, S-PLUS uses the same language, so I recommend books of this kind. Also, see UCLA's Statistical Computing page for online resources.

沒有留言:
標籤：

Stage 3 Statistical Inference

Statistical Inference (optional):
Build upon stage 2 statistics, you will learn here the Cramer-Rao bound, uniform minimum variance unbiased estimators, Neyman-Pearson theory, Bayesian inference, basic bootstrap and robustness, introductory non-parametric (including the sign test, Wilcoxon signed rank test, McNemar's chi-square test, Wald-Wolfowitz runs test, Mann-Whitney U test, Kolmogorov-Smirnov two-samples test, Kruskal-Wallis analysis of ranks, Spearman's R and Kendal's Tau), etc. Note a few texts say they are aimed at graduated level, partly because people from other fields only learn this material in graduated schools, as a result the texts have to set up basic probability and statistics (you have done it, stage 2 stuff) for them.
沒有留言:
標籤：

Stage 3 Mathematical Modelling

Mathematical Modelling (optional):
"All models are wrong, some are useful." George Box. Using mathematics to model real world phenomena is useful, but not the main concern of pure mathematician.

沒有留言:
標籤：

Stage 3 Introductory Topology

Introductory Topology:
Topology is an extension of geometry and built on set theory, it's about continuity. There is a joke that topologists are those who cannot distinguish between a doughnut and a coffee or tea cup. Wikipedia has a [.gif] to demonstrate this. You may have studied open sets, open neighbourhood, interior, closure, boundary, basis, continuity, compactness and connectedness etc. in the analysis course, move on to homemorphism, homotopy and fundamental group.

As in abstract algebra and analysis, several selected books or notes cover more than enough. You may leave a few sections to study at the next stage.

沒有留言:
標籤：

Stage 3 Differential Geometry

Differential Geometry:
You may wonder, geometry takes up a large portion in high school mathematics, why isn't there any geometry course in the first two stages? In fact, geometry is kind of imbedded in stage two calculus (several variables) and linear algebra courses, they are usually assumed and will be used for this course. Here differential calculus is used to study geometry. Key things to study: multilinear algebra, curvature and torsion, Serret-Frenet equation, fundamental theorem of curves, Poincaré Index theorem (plane and surface), exterior calculus, Gauss' theorema egregium, geodesics, Gauss-Bonnet theorem.

Moving frame, due to Élie Cartan, is an approach to geometry of surface. According to my lecturer John Steele, it is "computationally the easiest, notationally the neatest, aesthetically the best, makes the definitions more natural and the proofs of the two major theorems easier. The moving frame method also points the way towards several important ideas in modern differential geometry and theoretical physics. The downside (if there is one) is the reliance on exterior calculus of differential forms." O'Neill, for example, uses this approach and he manages to prove Gauss' theorema egregium in half page, see p.281.

In general:

Tensor Analysis and Manifolds:

Differential Forms:

沒有留言:
標籤：

Stage 3 Introductory Number Theory

Introductory Number Theory:
Gauss once said "mathematics is the queen of sciences and number theory the queen of mathematics." G..H. Hardy once said in A Mathematician's Apology that, "If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and less mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean." He wrote it on November 1940. It turned out coding theory, where algebra and number theory are applied, played a key role in the World War II.

沒有留言:
標籤：

Stage 3 Abstract Algebra

Abstract Algebra:
One of the beautiful branches of mathematics. You will see how we study symmetry. Topics to be covered are basic group, ring and field theory.

Only discuss Groups:

Rings and Fields:

In general:

沒有留言:
標籤：

Stage 3 Introductory Analysis

Introductory Analysis:
At this stage you should have learnt limits, continuity and convergence, over the reals. These are central concepts of calculus in both one and several variables. These can be generalised. You will study the idea of several spaces (metric space, topological space, Banach, Hilbert), compactness, connectedness, linear operators, elementary Lebesgue theory and measure theory, etc.

Contents of analysis courses varies almost everywhere (a.e.). As you may see, the term analysis appear in the first and second year calculus to forth year analysis. Kolmogorov and Fomin's text fits this stage well, they cover most of the above topics. Apostol, Marsden and Hoffman, and Simmons make good references. Several books entitled analysis are indeed advanced calculus (stage two calculus) or not aimed for mathematicians. Stay away from Mathematical analysis for business and finance or the like.

Only for metric spaces and topological spaces (also consult the list of Topology below):

Also deals with Lebesgue theory, Banach spaces and Hilbert spaces, etc:

沒有留言:
標籤：

Stage 3

At this stage, you begin to learn modern Pure Mathematics. Yup, this is the beginning. Focus on the proof. Don't expect you can solve a problem by plugging numbers into formulae. It often happens in advanced text that the author skips some steps in a proof or calculation, while elementary text gives detail explanations. But as G.F. Simmons has said, '[t]he serious student will train himself to look for gaps in proofs, and should regard them as tacit invitations to do a little thinking on his own.' Moreover, '[i]t is a basic principle in the study of mathematics, and one too seldom emphasized, that a proof is not really understood until the stage is reached at which one can grasp it as a whole and see it as a single idea. In achieving this end, much more is necessary than merely following the individual steps in the reasoning. This is only the beginning. A proof should be chewed, swallowed, and digested, and this process of assimilation should not be abandoned until it yields a full comprehension of the overall pattern of though.'

Also get maturity and learn how to write mathematics, by reading good books (?). I don't think Gauss or Galois can get full mark in mathematics assignments these days, for 1) something obvious to them may not be obvious to everyone else, and 2) they don't bother to explain to others. The point is, every steps in a proof should be logically related and everyone can go easily from the previous step to the next. This is not the case for non-pure mathematics. If you insist, I can show you a copy of full mark statistics assignment consists work of mere calculation and computation. (Of course not done by me!)

I had tried to list more books so that you can compare and choose one or two that fit you. Despite that most of following courses are 'pure', they can be applied to other graduated level science subjects. In the analysis class, Ian Doust even showed us an article in the Econometrica (Hildenbrad W. and Metrens J.F. Upper Hemi-continuity of the Equilibrium-set Correspondence for Pure Exchange Economies, Vol. 40, No. 1) talks about liminf, limsup, measure, weak topology and stuffs like that. If one insists that the pure stuff is useless, one is just ignorant, I don't bother to argue with those people anymore.

沒有留言:
標籤：