It is proved that in any pointed category with pullbacks, coequalizers and regular epi-mono factorizations, the class of regular epimorphisms is stable under pullback along the so-called balanced effective descent morphisms. Here “balanced” can be omitted if the category is additive. A balanced effective descent morphism is defined as an effective descent morphism p : E → B such that any subobj...

This paper describes an approach to the use of gradient descent search in genetic programming (GP) for object classification problems. Gradient descent search is introduced to the GP mechanism and is embedded into the genetic beam search, which allows the evolutionary learning process to globally follow the beam search and locally follow the gradient descent search. Two different methods, an on...

The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the Möbius function of intervals [1, π] in this poset, for any permutation π with at most one descent. We compute the Möbius function as a function of the number and positions of pairs of consecutive letters in π that are consecutive in value. As a result of this we show that the Möbius function is un...

!! Ui // X The modules Mi are “restrictions“ or pullbacks of M along Ui → X, and are compatible in that if Mi,j is the pullback of Mi over Ui along Ui ×X Uj → Ui, then we have isomorphisms Mi,j ∼= Mj,i that satisfy a coherence condition known as the cocycle condition. We say that modules over affine schemes have the property of descent for a family {Ui → X} if any family of compatible modules M...

The steepest descent method for large linear systems is well-known to often converge very slowly, with the number of iterations required being about the same as that obtained by utilizing a gradient descent method with the best constant step size and growing proportionally to the condition number. Faster gradient descent methods must occasionally resort to significantly larger step sizes, which...