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Zadeh fuzzy logic is normally introduced by assigning scalar membership degrees and postulating standard complement, minimum conjunction, and maximum disjunction. We derive this calculus from a strictly positive atomless normalized Boolean measure algebra. Equality of measure sends Boolean complement uniquely to $1-x$,...
The results provide a unified explanation for characteristic failure modes, such as local optima in alternating methods and greedy merge-order traps in hierarchical clustering, and show that clustering limitations reflect interacting computational and epistemic constraints, motivating a shift toward stability-aware obj...
This paper develops a unified theoretical framework showing that a broad family of clustering methods, including k-means, fuzzy c-means, kernel k-means, kernel FCM, and spectral clustering, can all be expressed as structured low-rank projectors acting on a signal-derived matrix. By formulating each method as an instanc...
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