Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. This dimension reduction procedure succeeds when it preserves certain geometric features of the set. The question is how large the embedding dimens...

Separoids capture the combinatorial structure which arises from the separations by hyperplanes of a family of convex sets in some Euclidian space. Furthermore, as we prove in this note, every abstract separoid S can be represented by a family of convex sets in the (|S| − 1)dimensional Euclidian space. The geometric dimension of the separoid is the minimum dimension where it can be represented a...

The language of gene expression displays topological symmetry. An important step during gene expression is the binding of transcriptional proteins to DNA promoters adjacent to a gene. Some proteins bind to many promoters in a genome, defining a regulon of genes wherein each promoter might vary in DNA sequence relative to the average consensus. Here we examine the linguistic organization of gene...

Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let F be a family of connected graphs Gn : F = (Gn)n ≥ 1 depending on n as follows: the order |V (G)| = φ(n) and lim n→∞ φ(n) = ∞. If there exists a constant C > 0 such that dim(Gn) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension, otherwise F has unbou...

For any family of measurable sets in a probability space, we show that either (i) the family has infinite Vapnik–Chervonenkis (VC) dimension or (ii) for every ε > 0 there is a finite partition π such the essential π -boundary of each set has measure at most ε. Immediate corollaries include the fact that a separable family with finite VC dimension has finite bracketing numbers, and satisfies uni...

We consider simple random walk on the family tree T of a nondegenerate supercritical Galton-Watson branching process and show that the resulting harmonic measure has a.s. strictly smaller Hausdorff dimension than that of the whole boundary of T . Concretely, this implies that an exponentially small fraction of the nth level of T carries most of the harmonic measure. First order asymptotics for ...

All genes on the human Y-chromosome were studied using fractal dimension and Shannon entropy. Clear outlier clusters were identified. Among these were 6 sequences that have since been withdrawn as CDSs and 1 additional sequence that is not in the current assembly. A methodology for ranking the sequences based on deviation from average values of FD and SE was developed. The group of sequences sc...