We have considered a simple word game called the word-morph. After making our participants play a stipulated number of word-morph games, we have analyzed the experimental data. We have given a detailed analysis of the learning involved in solving this word game. We propose that people are inclined to learn landmarks when they are asked to navigate from a source to a destination. We note that these landmarks are nodes that have high closeness-centrality ranking.

Given a sequence {Z d } d∈N of smooth and compact hypersurfaces in R n−1 , we prove that (up to extracting subsequences) there exists a regular definable hypersurface Γ ⊂ RP n such that each manifold Z d appears as a component of the zero set on Γ of some polynomial of degree d. (This is in sharp contrast with the case when Γ is algebraic, where for example the homological complexity of the zero set of a polynomial p on Γ is bounded by a polynomial in deg(p).)More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, definable hypersurface Γ ⊂ RP n containing a subset D homeomorphic to a disk, and a family of polynomials {pm} m∈N of degree degThis says that, up to extracting subsequences, the intersection of Γ with a hypersurface of degree d can be as complicated as we want. We call these "pathological examples".In particular, we show that for every 0 ≤ k ≤ n − 2 and every sequence of natural numbers a = {a d } d∈N there is a regular, compact and definable hypersurface Γ ⊂ RP n , a subsequence {a dm } m∈N and homogeneous polynomials {pm} m∈N of degree deg(pm) = dm such that:(Here b k denotes the k-th Betti number.) This generalizes a result of Gwoździewicz, Kurdyka and Parusiński [13].On the other hand, for a given definable Γ we show that the Fubini-Study measure, in the gaussian space of polynomials of degree d, of the set Σ dm,a,Γ of polynomials verifying () is positive, but there exists a contant c Γ such that this measure can be bounded by:This shows that the set of "pathological examples" has "small" measure (the faster a grows, the smaller the measure and pathologies are therefore rare). In fact we show that given Γ, for most polynomials a Bézout-type bound holds for the intersection Γ ∩ Z(p): for every 0 ≤ k ≤ n − 2 and t > 0:P {b k (Γ ∩ Z(p)) ≥ td n−1 } ≤ c Γ td n−1 2 . arXiv:1803.00539v1 [math.AG]

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