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This article investigates the probabilistic relationship between quantum classification of Boolean functions and their Hamming distance. By integrating concepts from quantum computing, information theory, and combinatorics, we explore how Hamming distance serves as a metric for analyzing deviations in function classification. Our extensive experimental results confirm that the Hamming distance is a pivotal metric for validating nearest neighbors in the process of classifying random functions. One of the significant conclusions we arrived is that the successful classification probability decreases monotonically with the Hamming distance. However, key exceptions were found in specific classes, revealing intra-class heterogeneity. We have established that these deviations are not random but are systemic and predictable. Furthermore, we were able to quantify these irregularities, turning potential errors into manageable phenomena. The most important novelty of this work is the demarcation, for the first time to the best of our knowledge, of precise Hamming distance intervals for the classification probability. These intervals bound the possible values the probability can assume, and provide a new foundational tool for probabilistic assessment in quantum classification. Practitioners can now endorse classification results with high certainty or dismiss them with confidence. This framework can significantly enhance any quantum classification algorithm’s reliability and decision-making capability.

The single-cell spatial transcriptomics (ST) data with cell type and spatial location, i.e., with C as cell type and as its spatial location, produced by recent biotechnologies, such as CosMx and Xenium, contain a huge amount of information about cancer tissue samples, thus have great potential for cancer research via detection of ST-Community which is defined as a collection of cells with distinct cell-type composition and similar neighboring patterns based on nearby cell-percentages. But for huge CosMx single-cell ST data, the existing clustering methods do not work well for st-community detection, and the commonly used kNN compositional data method shows lack of informative neighboring cell patterns. In this article, we propose a novel and more informative disk compositional data (DCD) method for single-cell ST data, which identifies neighboring patterns of each cell via taking into account of ST data features from recent new technologies. After initial processing single-cell ST data into the DCD matrix, an innovative DCD-TMHC computation method for st-community detection is proposed here. Extensive simulation studies and the analysis of CosMx breast cancer data, which is an example of single-cell ST dataset, clearly show that our proposed DCD-TMHC computation method is superior to other existing methods. Based on the st-communities detected for CosMx breast cancer data, the logistic regression analysis results demonstrate that the proposed DCD-TMHC computation method produces better interpretable and superior outcomes, especially in terms of assessment for different cancer categories. These suggest that our proposed novel and informative DCD-TMHC computation method here will be helpful and have an impact on future cancer research based on single-cell ST data, which can improve cancer diagnosis and monitor cancer treatment progress.

This study investigates the derivation of optimal repeated measurement designs of two treatments, five periods, and n experimental units for carryover effects. The optimal designs are determined for cases where n = 0, 1 (mod 2). The adopted optimality criterion focuses on minimizing the variance of the estimated carryover effect, thereby ensuring maximum precision in parameter estimation and design efficiency. The results presented here extend and complement earlier research of Chalikias and Kounias on optimal two-treatment repeated measurement designs for a smaller number of periods, and are closely related to the more recent findings on optimal designs for direct effects. Overall, the present work contributes to the theoretical framework of optimal design methodology by providing new insights into the structure and efficiency of repeated measurement designs, particularly in the presence of carryover effects, and sets the ground for future extensions incorporating treatment–period interactions.