Abstract. We systematize the study of reflection positivity in statistical mechanical models, and thereby two techniques in the theory of phase transitions: the method of infrared bounds and the chessboard method of estimating contour probabilities in Peierls arguments. We illustrate the ideas by applying them to models with long range interactions in one and two dimensions. Additional applications are discussed in a second paper.

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In this paper we identify conditions under which a true generator does or does not exist for an empirically observed Markov transition matrix. We show how to search for valid generators and choose the "correct" one that is the most compatible with bond rating behaviors. We also show how to obtain an approximate generator when a true generator does not exist. We give illustrations using credit rating transition matrices published by Moody's and by Standard and Poor's.

This paper discusses techniques for solving discrete optimization problems using quantum annealing. Practical issues likely to affect the computation include precision limitations, finite temperature, bounded energy range, sparse connectivity, and small numbers of qubits. To address these concerns we propose a way of finding energy representations with large classical gaps between ground and first excited states, efficient algorithms for mapping non-compatible Ising models into the hardware, and the use of decomposition methods for problems that are too large to fit in hardware. We validate the approach by describing experiments with D-Wave quantum hardware for low density parity check decoding with up to 1000 variables.

FMRI data acquisition under naturalistic and continuous stimuli (e.g., watching a video or listening to music) has become popular recently due to the fact that it entails less manipulation and more realistic/complex contexts involved in the task, compared to the conventional task-based experimental designs. The synchronization or response similarities among subjects are typically measured through inter-subject correlation (ISC) between any pair of subjects. At the group level, summarizing the collection of ISC values is complicated by their intercorrelations, which necessarily lead to the violation of independence assumed in typical parametric approaches such as Student’s t-test. Nonparametric methods, such as bootstrapping and permutation testing, have previously been adopted for testing purposes by resampling the time series of each subject, but the quantitative validity of these specific approaches in terms of controllability of false positive rate (FPR) has never been explored before. Here we survey the methods of ISC group analysis that have been employed in the literature, and discuss the issues involved in those methods. We then propose less computationally intensive nonparametric methods that can be performed at the group level (for both one- and two-sample analyses), as compared to the popular method of circularly shifting the EPI time series at the individual level. As part of the new approaches, subject-wise (SW) resampling is adopted instead of element-wise (EW) resampling, so that exchangeability and independence assumptions are satisfied, and the patterned correlation structure among the ISC values can be more accurately captured. We examine the FPR controllability and power achievement of all the methods through simulations, as well as their performance when applied to a real experimental dataset. The new methodologies are shown to be both efficient and robust, and they have been implemented into an open source program, 3dNPT, in AFNI (http://afni.nimh.nih.gov).