To apply genetic programming (GP) to evolve λ-expressions, we devised an encoding scheme that encodes λ-expressions into trees. This encoding has closure property, i.e., any combination of terminal and non-terminal symbols forms a valid λ-expression. We applied this encoding to a simple symbolic regression problem over Church numerals and the objective function f(x) = 2x was successfully obtain...

For vertices x and y in a connected graph G, the detour distance D(x, y) is the length of a longest x− y path in G. An x− y path of length D(x, y) is an x − y detour. The closed detour interval ID[x, y] consists of x, y, and all vertices lying on some x − y detour of G; while for S ⊆ V (G), ID[S] = ⋃ x,y∈S ID[x, y]. A set S of vertices is a detour convex set if ID[S] = S. The detour convex hull...

In the paper a theoretical analysis is given for the smallest ball that covers a finite number of points p1, p2, · · · , pN ∈ R . Several fundamental properties of the smallest enclosing ball are described and proved. Particularly, it is proved that the k-circumscribing enclosing ball with smallest k is the smallest enclosing ball, which dramatically reduces a possible large number of computati...

In a seminal paper of Charikar et al. on the smallest grammar problem, the authors derive upper and lower bounds on the approximation ratios for several grammar-based compressors, but in all cases there is a gap between the lower and upper bound. Here we close the gaps for LZ78 and BISECTION by showing that the approximation ratio of LZ78 is Θ((n/ logn)), whereas the approximation ratio of BISE...

A property of n-vertex graphs is called evasive if every algorithm testing this property by asking questions of the form “is there an edge between vertices u and v” requires, in the worst case, to ask about all pairs of vertices. Most “natural” graph properties are either evasive or conjectured to be such, and of the few examples of nontrivial nonevasive properties scattered in the literature t...

A Lipschitz map f between the metric spaces X and Y is called a Lipschitz quotient map if there is a C > 0 (the smallest such C, the co-Lipschitz constant, is denoted coLip(f), while Lip(f) denotes the Lipschitz constant of f) so that for every x ∈ X and r > 0, fBX(x, r) ⊃ BY (f(x), r/C). Thus Lipschitz quotient maps are surjective maps which by definition have the property ensured by the open ...

We describe our efforts to study dwarf galaxies with active nuclei, whose black holes, with masses ≤ 106 M⊙, provide the best current observational constraints on the mass distribution of primordial seed black holes. Although these low-mass galaxies do not necessarily contain classical bulges, Barth, Greene, & Ho (2005) show that their stellar velocity dispersions and black hole masses obey the...