Development of new algorithms in the area machine learning, especially clustering, comparative studies such as well testing according to software engineering principles requires availability labeled data sets. While standard benchmarks are made available, a broader range sets is necessary order avoid problem overfitting. In this context, theoretical works on axiomatization clustering algorithms...

In this paper, the Urysohn and completely Hausdorff axioms in general topology are generalized to L-topological spaces so as to be compatible with pointwise metrics. Some properties and characterizations are also derived

In the theory of classical matroids, there are several known equivalent axiomatic systems that define a matroid, which described as matroid cryptomorphisms . A q -matroid is -analogue where subspaces play role subsets in theory. this article we establish -matroids. doing so highlight difference between and its -analogue. We introduce comprehensive set axiom show them existing -matroid. These ax...

In the paper we prove that axiom CPA prism, which follows from the Covering Property Axiom CPA and holds in the iterated perfect set model, implies that there exists a Hamel basis which is a union of less than continuum many pairwise disjoint perfect sets. We will also give two consequences of this last fact. 1 The result and its consequences In this paper we will use standard set theoretic ter...

This extended abstract gives an introduction into the development of partial correctness logics for programming languages speci ed by evolving algebras. A partial correctness logic is a programming logic that allows to prove program properties of the form: \whenever program point P is reached during execution, assertion A is true". We derive a basic axiom (schema) from an evolving algebra and u...

The paper investigates the strength of the Anti-Foundation Axiom, AFA, on the basis of Kripke-Platek set theory without Foundation. It is shown that the addition of AFA considerably increases the proof theoretic strength. 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in diiering elds such as mathematical logic, computer science, artiicial intelligenc...

We prove the topological (or combinatorial) rigidity property for real polynomials with all critical points real and nondegenerate, which completes the last step in solving the density of Axiom A conjecture in real one-dimensional dynamics.