In the application of fuzzy reasoning, researchers usually choose the membership function optionally in some degree. Even though the membership functions may be different for the same concept, they can generally get the same or approximate results. The robustness of the membership function optionally chosen has brought many researchers’ attention. At present, many researchers pay attention to t...

We consider closed topological 4-manifolds M with universal cover S × S and Euler characteristic χ(M) = 1. All such manifolds with π = π1(M) ∼= Z/4 are homotopy equivalent. In this case, we show that there are four homeomorphism types, and propose a candidate for a smooth example which is not homeomorphic to the geometric quotient. If π ∼= Z/2×Z/2, we show that there are three homotopy types (a...

Some ball-quotient orbifolds are related by covering maps. We exploit these coverings to find infinitely many orbifolds on P uniformized by the complex 2-ball B2 and some orbifolds over K3 surfaces uniformized by B2. We also give, along with infinitely many reducible examples, an infinite series of irreducible curves along which P is uniformized by the product of 1-balls B1× B1.

This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M and α ≥ 1, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embedings int...

The purpose of this paper is to draw attention to the widespread occurrence of quotient spaces in statistical work. Quotient spaces are intrinsic to probability distributions, residuals and interaction in linear models, covariance functions and variograms of stochastic processes, etc. The theme is that explicit recognition of the quotient space can offer surprising conceptual simplification. Th...

Let the special linear group G := SL 2 act regularly on a Q-factorial variety X. Consider a maximal torus T ⊂ G and its normalizer N ⊂ G. We prove: If U ⊂ X is a maximal open N-invariant subset admitting a good quotient U → U/ /N with a divisorial quotient space, then the intersection W (U) of all translates g·U is open in X and admits a good quotient W (U) → W (U)/ /G with a divisorial quotien...