Bjørn Kjos-Hanssen

Statistical Inference

Math 472

Textbook: An introduction to mathematical statistics and its applications (Larsen and Marx), 5th edition
Times: TuTh 9-10:15

Homework list
3.8: 1, 2, 6 (Transforming and combining random variables)
3.10: 8, 12, 14 (Order statistics)
4.6: 1, 4, 5 (Gamma distribution)

5.2: 2, 10, 14 (Method of maximum likelihood)

5.3: 1, 13, 22 (Interval estimation)
5.4: 14, 16, 17
5.5: 2, 3, 7
5.6: 1, 5, 6
5.7: 3, 4, 5
5.8: 3, 5, 7
7.3: 4, 6, 8
7.4: 1, 3, 25
7.5: 7, 8, 14
9.2: 1, 5, 16
9.3: 1, 4, 10
9.4: 3, 6, 8
11.2: 10, 22, 29
11.3: 11, 18, 19
11.4: 1, 5,14
11.5: 1, 8, 9

Each homework set is worth the same amount. The top 10 of 12 homeworks count 2% of the course each. The best out of two midterms counts 40%. The final counts 40%.

Transition to Advanced Mathematics

Math 321

Textbook: Numbers, groups, and codes (Humphreys and Prest), 2nd edition
Times: TuTh 12-1:15

Each homework set is worth the same amount. The top 10 out of 12 homeworks count 2% of the course each for a total of 20% for homework. The best out of two midterms counts 40%. The final counts 40%.

1.1: 3, 6 (Greatest common divisors)
1.2: 3, 7 (Induction)
1.3: 6, 7, 8 (Unique factorization)

1.5: 1, 4 (Solving linear congruences)

1.6: 1, 4, 7, 10, 12
2.1: 3, 7, 8
2.2: 7, 8, 9, 11
2.3: 2, 3, 5, 10
2.4: 2, 4
3.1: 2, 4
3.2: 1
3.3: 3
4.1: 2, 5
4.2: 3, 4, 5, 8, 12
4.3: 3, 4, 7
4.4: 3, 4, 5, 6, 8, 14
5.1: 1, 2, 6, 7
5.2: 1, 3
5.3: 2, 3, 5, 8
5.4: 2, 3, 5

A strong law of computationally weak subsets

A strong law of computationally weak subsets

Journal of Mathematical Logic 11 (2011) no. 1, 1—10.
DOI: 10.1142/S0219061311000980
Electronic Colloquium on Computational Complexity, Report No. 150 (2010).

This paper establishes that each 2-random set has an infinite subset computing no 1-random set. It is perhaps the main result obtained under grant NSF DMS-0901020 (2009-2013).
Joseph S. Miller has established a strengthening of this result replacing 2-random by 1-random, but I believe the proof of that result has not yet been written up.

Kolmogorov complexity and the recursion theorem

Kolmogorov complexity and the recursion theorem (with Wolfgang Merkle and Frank Stephan). Transactions of the American Mathematical Society363 (2011) no. 10, 5465—5480.
arXiv:0901.3933. Preliminary version in STACS 2006, Lecture Notes in Computer Science,
vol. 3884, Springer, Berlin, 2006, pp. 149—161.

The groundwork for this paper was laid in Berkeley in 2004 while Merkle and I were both stopping by there. The paper fits in a long line of results saying “DNR is equivalent to…”, in this case to computing a real whose initial segment complexity is sufficiently high.

Kolmogorov complexity and strong approximation of Brownian motion

Kolmogorov complexity and strong approximation of Brownian motion
(with Tamás Szabados). Proceedings of the American Mathematical Society 139 (2011) no. 9, 3307—3316.

In this email-collaboration paper we improved a bound from Asarin (Kolmogorov’s PhD student) and Pokrovskii’s paper on Kolmogorov complexity and Brownian motion.

The probability distribution as a computational resource for randomness testing

The probability distribution as a computational resource for randomness testing
Journal of Logic and Analysis 2 (2010), no. 10, 1—13.

This paper grew out of my assignment to teach Math 472, Statistical Inference, in Spring 2009 at UH. By analyzing the proof of the law of large numbers and some other work I showed that Hippocratic and Galenic randomness coincide for Bernoulli measures. [There has been follow-up work to this paper which it would make sense to describe here.]

Higher Kurtz randomness

Higher Kurtz randomness
(with André Nies, Frank Stephan, and Liang Yu). Annals of Pure and Applied Logic 161 (2010), no. 10, 1280—1290.

I visited Liang Yu at Nanjing University, China in 2008 and 2009. The collaboration for this paper however was later done by email plus a strategy meeting in Marseille in 2009.

Lattice initial segments of the hyperdegrees

Lattice initial segments of the hyperdegrees
(with Richard A. Shore). Journal of Symbolic Logic 75 (2010), no. 1, 103—130.

I was a postdoc with Shore during the academic year 2006-2007 at Cornell, and this paper was the result of our joint work there. It concerns hyperarithmetical reducibility and its induced partial ordering of the reals.

Superhighness

Superhighness
(with André Nies). Notre Dame Journal of Formal Logic 50 (2009), no. 4, 445—452.

This paper written on Maui has since been superseded in at least two ways.

Finding paths through narrow and wide trees

Finding paths through narrow and wide trees
(with Stephen E. Binns). Journal of Symbolic Logic 74 (2009), no. 1, 349—360.

This paper was written at UConn in 2006. I believe there has not been follow-up work on its topic yet.

• Statistical Inference
• Transition to Advanced Mathematics
• A strong law of computationally weak subsets

• automata
• Brownian motion
• NSF DMS-0901020
• publications
• randomness
• research
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