This study proposes modular arithmetic erasure channels (MAECs), a novel class of erasure-like channels with an input alphabet that need not be binary. This class contains the binary erasure channel (BEC) and some other known erasure-like channels as special cases. For MAECs, we provide recursive formulas of Arıkan-like polar transform to simulate channel polarization. In other words, we show that the synthetic channels of MAECs are equivalent to other MAECs. This is a generalization of well-known recursive formulas of the polar transform for BECs. Using our recursive formulas, we also show that a recursive application of the polar transform for MAECs results in multilevel channel polarization, which is an asymptotic phenomenon that is characteristic of non-binary polar codes. Specifically, we establish a method to calculate the limiting proportions of the partially noiseless and noisy channels that are generated as a result of multilevel channel polarization for MAECs. In the particular case of MAECs, this calculation method solves an open problem posed by Nasser (2017) in the study of non-binary polar codes.

Polar transforms are central operations in the study of polar codes. This paper examines polar transforms for non-stationary memoryless sources on possibly infinite source alphabets. This is the first attempt of source polarization analysis over infinite alphabets. The source alphabet is defined to be a Polish group, and we handle the Arikan-style two-by-two polar transform based on the group. Defining erasure distributions based on the normal subgroup structure, we give recursive formulas of the polar transform for erasure distributions. We then show concrete examples of multilevel source polarization with countably infinite levels when the group is locally cyclic. We derive this result via elementary techniques in lattice theory.

This study examines multilevel channel polarization for a certain class of erasure channels with arbitrary input alphabet size. We derive limiting proportions of partially noiseless channels for such a class. One of the results of this study are proved by an argument of convergent sequences, inspired by Alsan and Telatar's simple proof of polarization [IEEE Transactions on Information Theory, vol. 62, no. 9, pp. 4873-4878, 2016], and without martingale convergence theorems for polarization process. Technical parts of this study can be found in the arXiv at [https://arxiv.org/abs/1801.04422].

For a probability vector (p, 1 - p), we assume that ß = 1/p is a reduced quadratic irrational. Then, we show that the coding function for the asymmetric binary systems (ABS) determines a factor of the subshift arising from a substitution. For the case where ß is the golden mean, we give an explicit formula of the probability of states in the ABS.

We consider the embedding problem for shifts of finite type (SFT) into the Fibonacci-Dyck shift. First, we find the reflection principle does not hold for the path in the Fibonacci-Dyck shift. Then, we obtain the limit of a sequence of topological entropies of a class of irreducible (d - 1)-step SFT of the Fibonacci-Dyck shift, where d(≥ 2) are even integers. This limit provide one of the embedding conditions for the Fibonacci-Dyck shift. Finally, we generalize the result on computing the above limit of a sequence of topological entropies to the Markov-Dyck shift.

Large-scale conformational changes in proteins involve barrier-crossing transitions on the complex free energy surfaces of high-dimensional space. Such rare events cannot be efficiently captured by conventional molecular dynamics simulations. Here we show that, by combining the on-the-fly string method and the multi-state Bennett acceptance ratio (MBAR) method, the free energy profile of a conformational transition pathway in Escherichia coli adenylate kinase can be characterized in a high-dimensional space. The minimum free energy paths of the conformational transitions in adenylate kinase were explored by the on-the-fly string method in 20-dimensional space spanned by the 20 largest-amplitude principal modes, and the free energy and various kinds of average physical quantities along the pathways were successfully evaluated by the MBAR method. The influence of ligand binding on the pathways was characterized in terms of rigid-body motions of the lid-shaped ATP-binding domain (LID) and the AMP-binding (AMPbd) domains. It was found that the LID domain was able to partially close without the ligand, while the closure of the AMPbd domain required the ligand binding. The transition state ensemble of the ligand bound form was identified as those structures characterized by highly specific binding of the ligand to the AMPbd domain, and was validated by unrestrained MD simulations. It was also found that complete closure of the LID domain required the dehydration of solvents around the P-loop. These findings suggest that the interplay of the two different types of domain motion is an essential feature in the conformational transition of the enzyme.

In this paper, we consider asynchronous spread spectrum multiple access (SSMA) communication systems with spreading sequences of Markov chains. We show that, given phase shifts and time delays, the empirical mean of the multiple access interference (MAI) for such systems satisfies the conditional large deviations principle (LDP) with the convex, good rate function. By refinement of the large deviations analysis, we also obtain exact asymptotics of bit-error probabilities in such systems. Comparing theoretical expressions of bit-error probabilities with experimental results, we confirm that for too small numbers of users compared to the lengths of spreading sequences, the central limit asymptotics become invalid, but the large deviations asymptotics turn out to be relevant.

We consider the embedding problem for shifts of finite type into the golden-mean-Dyck shift. To this end, we study periodic points in the golden-mean-Dyck shift. We obtain the number of such periodic points of length n in the form of the trace of the n-th term of a sequence of the associated matrices. In view of this result, we also give simple examples of embeddings of shifts of finite type into the golden-mean-Dyck shift. Additionally, we generalize this result to the Markov-Dyck shift.

We have introduced the induced transformations associated with the interval algorithm. In view of the entropy of the induced transformations, the stopping time of the algorithm is evaluated. Sharp upper and lower bounds of the mean value of the stopping time are given for the case that the source and the target have uniform distributions.

In this report we consider asynchronous spread spectrum multiple access (SSMA) communication systems with spreading sequences of Markov chains. We show that, given phase shifts and time delays, the empirical mean of the multiple access interference (MAI) for such systems satisfies the conditional large deviation principle (LDP) with the convex good rate function. By refinement of the large deviations analysis, we obtain exact asymptotics of bit error probabilities in such systems.

We have recently shown that a class of ergodic maps with the equidistributivity property (or briefly EDP) can easily produce a sequence of independent and identically distributed (i.i.d.) binary random variables. In this note, it is shown that the Jacobian elliptic Chebyshev rational maps defined here have the EDP and the semi-group property like the Chebyshev polynomial maps. Simple methods are also discussed for generating sequences of i.i.d. random variables based on the Jacobian elliptic Chebyshev rational maps.

Sequences of random numbers generated by a Markov chain have been recently discussed as one of candidates of spreading sequences. We theoretically evaluate distributions of even and odd crosscorrelation values of sequences of binary random variables generated by Markov information sources.