We propose to extend the ontology of logical AI to include approximate objects, approximate predicates and approximate theories. Besides the ontology we treat the relations among different approximate theories of the same phenomena. Approximate predicates can’t have complete if-and-only-if definitions and usually don’t even have definite extensions. Some approximate concepts can be refined by l...

Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups: functions ψ : G → Ud such that Pr[ψ(xy) = ψ(x) ψ(y)] is large, or more generally Ex,y ‖ψ(xy)− ψ(x) ψ(y)‖22 is small, where x, y are uniformly random elements ...

Abstract Principal Component Analysis (PCA) is one of the most widely used data analysis methods in machine learning and AI. This manuscript focuses on mathematical foundation classical PCA its application to a small-sample-size scenario large dataset high-dimensional space scenario. In particular, we discuss simple method that can be approximate latter case. also help kernel or (KPCA) for larg...

the integration of an approximate model and a clonal selection algorithm (csa) is considered to one of the most effective ways to solve complex optimization design problems in engineering. in this study, first, a process for developing an approximate model and the principles of clonal selection algorithms were presented. then, a variotherm temperature injection mold used to produce a large liqu...

In this paper, within the framework of consistent approach recently introduced for approximate Lie symmetries differential equations, we consider Noether variational problems involving small terms. Then, state an theorem leading to construction conservation laws. Some illustrative applications are presented.