# Tag Info

0

Pinter's A Book of Abstract Algebra (Second Edition) is quite good.

0

You can find 'A Term of Commutative Algebra' by Allen Altman and Steven Kleiman (introduces Category Theory and has full solutions) which is a more in depth and updated version of 'Introduction to Commutative Algebra' by Atiyah, Macdonald. After that you could try Commutative Algebra I/II by Zariski, Samuel. Whilst simultaneously getting acquainted with ...

0

Start out with a sample path from a continuous Brownian motion that is positive. It is continuous but nowhere differentiable. Since it is continuous one may integrate it to obtain a differentiable map whose derivative is not differentiable (by the Fundamental Theorem of Calculus). Since it is positive it integral is an increasing function. Do this k times.

0

These notes seem brief and to the point: http://www.math.leidenuniv.nl/~edix/tag_2009/michiel_3.pdf

0

Milnor's book on K-theory, Princeton University Press

0

A book on this subject: "The Surprising Mathematics of Longest Increasing Subsequences", A book in progress by Dan Romik. You are interested in Section 1.17 (version 1.1 of the draft).

2

The length of the first row of a Plancherel-random Young diagram (with $n$ boxes) has the same distribution as the longest increasing subsequence of a random permutation $\pi$ in the symmetric group $S_n$ (hint: use Robinson-Schensted correspondence). By Markov's inequality, the probability that $\pi$ contains a decreasing sequence of length at least ...

2

John gave a (pretty standard) example as a comment: the surfaces $$\{(x,y,0):0<x<\pi, 0<y<1\}$$ and $$\{(\cos x,y,\sin x): 0<x<\pi, 0<y<1\}$$ are isometric in their intrinsic metrics, but are not extrinsically isometric.

3


0

It's not a great answer (I've not enough reputation to comment) but it explains the concept without much mathematical garnish: http://linbaba.wordpress.com/2010/06/02/doob-h-transforms/ A book (page 242) Markov chains and mixing times - David A. Levin, Yuval Peres, Elizabeth L. Wilmer

2

Check out: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations By Ratan Prakash Agarwal, Ravi P. Agarwal, V. Lakshmikantham Some more examples: Can a differential equation have non unique solutions? Search the web for notes with more examples.

1

It is not hard to see that it is as you say when $\mathcal A$ is a von Neumann algebra. But there are C$^*$-algebras with few to no projections, and so none of those masas can live there. Consider for instance $C^*_r(\mathbb F_2)$, the reduced C$^*$-algebra of the free group on two generators $\mathbb F=\langle a,b\rangle$. This algebra is known to be ...

1

The commonly used name is $j$-invariant. See here or here (you will have to read a little to realize that they are talking about the same thing!)

0

It seems that I was right in thinking that this is simply categorical logic; particularly, categorical proof theory. These notes look ideal. [Please do correct me if I'm wrong. Further information is welcome of course.]

1

Historical reference is Steenrod N. E. Cohomology invariants of mappings // Ann. Math. 1949. V. 50. P. 954—988.

1

Your argument is very interesting. I'm not sure how to best adapt it yet. The standard arguments would say that eigenfunctions in $L^{2}$ with different eigenvalues are orthogonal. Because $L^{2}[0,1]$ is separable (has a countable orthonormal basis), then there can be at, most, a countably-infinite number of eigenvalues. For non-singular Sturm-Liouville ...

1

To understand statistics from the very grassroots, I think the book Statistics, 4th edition by Freedman, Purves, Pisani will be very appropriate. The most basic book on Probability is An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition by Feller. And you are telling me that you are not also sure about Lebesgue-Stieltjes Integral. ...

1

I would suggest Ramanujan's collected works.

Top 50 recent answers are included