Conceptual Scaling is a useful standard tool in Formal Concept Analysis and beyond. Its mathematical theory, as elaborated the last chapter of FCA monograph, still has room for improvement. As it stands, even some basic definitions are flux. Our contribution was triggered by study concept lattices tree classifiers scaling methods used there. We extend notions, give precise them introduce dimens...

We show that an artin algebra Λ having at most three radical layers of infinite projective dimension has finite finitistic dimension, generalizing the known result for algebras with vanishing radical cube. We also give an equivalence between the finiteness of fin.dim.Λ and the finiteness of a given class of Λ-modules of infinite projective dimension.

Let ν be a finite Borel measure on [0, 1]. Consider the L-spectrum of ν: τν(q) = lim infn→∞−n logb P Q∈Gn ν(Q) q (q ≥ 0), where Gn is the set of b-adic cubes of generation n (b integer ≥ 2). Let qτ = inf{q : τν(q) = 0} and Hτ = τ ′ ν(q τ ). When ν is a mono-dimensional continuous measure of information dimension D, (qτ , Hτ ) = (1, D). When ν is purely discontinuous, its information dimension i...

This paper establishes an upper bound on the size of a concept class with given recursive teaching dimension (RTD, a teaching complexity parameter.) The upper bound coincides with Sauer’s well-known bound on classes with a fixed VC-dimension. Our result thus supports the recently emerging conjecture that the combinatorics of VC-dimension and those of teaching complexity are intrinsically interl...

Consider the problem of calculating the fractal dimension of a set X consisting of all infinite sequences S over a finite alphabet Σ that satisfy some given condition P on the asymptotic frequencies with which various symbols from Σ appear in S. Solutions to this problem are known in cases where (i) the fractal dimension is classical (Hausdorff or packing dimension), or (ii) the fractal dimensi...

Let X = {X(t), t ∈ RN} be a Gaussian random field with values in R defined by X(t) = ( X1(t), . . . , Xd(t) ) , ∀ t ∈ R , where X1, . . . , Xd are independent copies of a centered Gaussian random field X0. Under certain general conditions, Xiao (2007a) defined an upper index α∗ and a lower index α∗ for X0 and showed that the Hausdorff dimensions of the range X ( [0, 1] ) and graph GrX ( [0, 1] ...

This paper is concerned with various combinatorial parameters of classes that can be learned from a small set of examples. We show that the recursive teaching dimension, recently introduced by Zilles et al. (2008), is strongly connected to known complexity notions in machine learning, e.g., the self-directed learning complexity and the VC-dimension. To the best of our knowledge these are the fi...

We calculate the box-counting dimension of the limit set of a general geometrically finite Kleinian group. Using the “global measure formula” for the Patterson measure and using an estimate on the horoball counting function we show that the Hausdorff dimension of the limit set is equal to both: the box-counting dimension and packing dimension of the limit set. Thus, by a result of Sullivan, we ...

We construct a Π1-class X that has classical packing dimension 0 and effective packing dimension 1. This implies that, unlike in the case of effective Hausdorff dimension, there is no natural correspondence principle (as defined by Lutz) for effective packing dimension. We also examine the relationship between upper box dimension and packing dimension for Π1-classes.