Independent sets of a given size and structure in the hypercube

We determine the asymptotics of the number of independent sets of size $\lfloor \beta 2^{d-1} \rfloor$ in the discrete hypercube $Q_d = \{0,1\}^d$ for any fixed $\beta \in (0,1)$ as $d \to \infty$ , extending a result of Galvin for $\beta \in (1-1/\sqrt{2},1)$ . Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in $Q_d$ drawn according to the hard-core model at any fixed fugacity $\lambda>0$ . In proving these results we develop several general tools for performing combinatorial enumeration using polymer models and the cluster expansion from statistical physics along with local central limit theorems.

Standard mark table and standard USCI-CF table of the point group D6h. Unification of candidates derived from different D6h-skeletons

Combined-permutation representations (CPRs) for characterizing -skeletons (a benzene skeleton, a Haworth-projected skeleton, a superphane skeleton, and a coronene skeleton) are constructed by starting from respective sets of generators, where the permutation of each generator is combined with a mirror-permutation of 2-cycle to treat both achiral and chiral substituents under the GAP system. Thereby, the CPR of degree 8 (= 6 + 2) for the benzene skeleton, the CPR of degree 14 (= 12 + 2) for the Haworth-projected skeleton, the CPR of degree 14 (= 12 + 2) for the superphane skeleton, the CPR of degree 14 (= 12 + 2) for the coronene skeleton are generated to give primary mark tables (tables of marks) based on these CPRs. These primary mark tables generated by the GAP system are different in the sequence of subgroups from each other, although they stem from the same point group . They are unified into a single standard mark table by means of a newly-devised GAP function MarkTableforUSCI. Moreover, another newly-devised GAP function constructUSCITable is employed to construct a standard USCI-CF (unit-subduced-cycle-index-with-chirality-fittingness) table concordantly. After a set of PCI-CFs (partial cycle indices with chirality fittingness) is calculated for each skeleton, symmetry-itemized combinatorial enumeration is conducted by means of the PCI method of Fujita’s USCI approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991).

Enhanced Hyperbox Classifier Model for Nanomaterial Discovery

Machine learning tools can be applied to peptide-mediated biomineralization, which is an emerging biomimetic technique of creating functional nanomaterials. In particular, they can be used for the discovery of biomineralization peptides, which currently relies on combinatorial enumeration approaches. In this work, an enhanced hyperbox classifier is developed which can predict if a given peptide sequence has a strong or weak binding affinity towards a gold surface. A mixed-integer linear program is formulated to generate the rule-based classification model. The classifier is optimized to account for false positives and false negatives, and clearly articulates how the classification decision is made. This feature makes the decision-making process transparent, and the results easy to interpret for decision support. The method developed can help accelerate the discovery of more biomineralization peptide sequences, which may expand the utility of peptide-mediated biomineralization as a means for nanomaterial synthesis.

Two Undergraduate Students' Reinvention of the Multiplication Principle

The multiplication principle (MP) is a fundamental aspect of combinatorial enumeration, serving as an effective tool for solving counting problems and underlying many key combinatorial formulas. In this study, the authors used guided reinvention to investigate 2 undergraduate students' reasoning about the MP, and they sought to answer the following research questions: How do students come to understand and make sense of the MP? Specifically, while a pair of students reinvented a statement of the MP, how did they attend to and reason about key mathematical features of the MP? The students participated in a paired 8-session teaching experiment during which they progressed from a nascent to a sophisticated statement of the MP. Two key mathematical features emerged for the students through this process, including independence and distinct composite outcomes, and we discuss ways in which these ideas informed the students' reinvention of the statement. In addition, we present potential implications and directions for future research.

New approaches for assessing the activities of operators of complex technical systems

Presented are new approaches for supporting the outcome grading for activities of operators of complex technical systems, which are based on comparisons of current exercises with the activity database patterns in both the wavelet representation metric associated with time series of activity parameters and the likelihood metric of eigenvalue trajectories for these parameters transforms as well as on probabilistic assessments of skill class recognition using sample distribution functions of exercise distances to cluster centers in a scaling space and Bayesian likelihood estimations with the aid of probabilistic profile of staying in activity parameter ranges. These techniques have demonstrated the capabilities of recognizing sets of abnormal exercises and detection of parameters characterizing operator mistakes to reveal the causes of abnormality. The techniques in question overcome limitations of existing methods and provide advantages over manual data analysis since they greatly reduce the combinatorial enumeration of the options considered.