The $3x 1$ problem, or Collatz problem, concerns the following seemingly innocent arithmetic procedure applied to integers: If an integer $x$ is odd then ""multiply by three and add one"", while if it is even then ""divide by two"". The $3x 1$ problem asks whether, starting from any positive integer, repeating this procedure over and over will eventually reach the number 1. Despite its simple appearance, this problem is unsolved. Generalizations of the problem are known to be undecidable, and the problem itself is believed to be extraordinarily difficult. This book reports on what is known on this problem. It consists of a collection of papers, which can be read independently of each other. The book begins with two introductory papers, one giving an overview and current status, and the second giving history and basic results on the problem. These are followed by three survey papers on the problem, relating it to number theory and dynamical systems, to Markov chains and ergodic theory, and to logic and the theory of computation. The next paper presents results on probabilistic models for behavior of the iteration. This is followed by a paper giving the latest computational results on the problem, which verify its truth for $x
MR3513873 Palle E. T. Jorgensen, Karl-Hermann Neeb, and Gestur Olafsson, ́ Reflection positivity on real intervals, Semigroup Forum 96 (2018), no. 1, 31–48. MR3750345 Palle E. T. Jorgensen and Gestur Olafsson, ́ Unitary representations ...
本书共有二十八个书写键盘练习,六个分析练习。根据和声学的功能原则与声部的运动原则以及经典音乐作品的实践,作者对练习中的和弦序进以及声部进行都提出了最佳方案。
本书从经典傅里叶分析的清晰表述入手, 旨在用一个具体的构架展示调和分析中心思想. 作者在确立这些思想之后, 转向扩展和分析, 使之远远超越圆群的范围, ...
Lp Harmonic Analysis on SL (2, R)
Explores some basic roles of Lie groups in linear analysis, with particular emphasis on the generalizations of the Fourier transform and the study of partial differential equations.
Wavelets through a looking glass. Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc., Boston, MA, 2002. The world of the spectrum. Lawrence Baggett, Palle Jorgensen, Kathy Merrill, and Judith Packer.
Then the representations of OA associated to these monic systems are equivalent if and only if the measures m and m are equivalent, ... Palle E. T. Jorgensen, and Vasyl Ostrovsky ̆ı, Representation theory and numerical AF-invariants.
Mathematical Surveys and Monographs
Convergence of wavelet expansions 1.8. Wavelet frames Chapter 2. Multivariate Wavelets 2.1. Separable MRA 2.2. Matrix dilation 2.3. Nonseparable MRAs 2.4. Construction of refinable functions 2.5. Conditions of biorthogonality 2.6.
The Scope and History of Commutative and Noncommutative Harmonic Analysis