We introduce the notion of an adequate moduli space. The theory of adequate moduli spaces provides a framework for studying algebraic spaces which geometrically approximate algebraic stacks with reductive stabilizers in characteristic p. The definition of an adequate moduli space generalizes the existing notion of a good moduli space to characteristic p (and mixed characteristic). The most impo...

In this article we formalize a quotient module of Z-module and a vector space constructed by the quotient module. We formally prove that for a Z-module V and a prime number p, a quotient module V/pV has the structure of a vector space over Fp. Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [14]. Some...

Values of some characteristic parameters of confined explosions for LPG (Liquefied Petroleum Gas)-air quiescent mixtures are examined in correlation with the initial pressure of fuel-air mixture and fuel/air ratio. The explosion pressures, the explosion times and the maximum rates of pressure rise were obtained from experiments made at ambient initial temperature in two closed vessels with cent...

A general framework is laid out for principal component analysis (PCA) on quotient spaces that result from an isometric Lie group action on a complete Riemannian manifold. If the quotient is a manifold, geodesics on the quotient can be lifted to horizontal geodesics on the original manifold. Thus, PCA on a manifold quotient can be pulled back to the original manifold. In general, however, the q...

Abstract: Let R be an open equivalence relation on a topological space E. We define on E a new equivalence relation ̃︀ R by x̃︀ Ry if the closure of the R-trajectory of x is equal to the closure of the R-trajectory of y. The quotient space E/̃︀ R is called the trajectory class space. In this paper, we show that the space E/̃︀ R is a simple model of the quotient space E/R. This model can provide a f...

We prove that the dual or any quotient of a separable reflexive Banach space with the unconditional tree property has the unconditional tree property. This is used to prove that a separable reflexive Banach space with the unconditional tree property embeds into a reflexive Banach space with an unconditional basis. This solves several long standing open problems. In particular, it yields that a ...

We generalize the concept of global moment maps to local moment maps whose different branches are labelled by the elements of the fundamental group of the underlying symplectic manifold. These branches can be smoothly glued together by employing fundamental-group-valued C̆ech cocycles on the phase space. In the course of this work we prove a couple of theorems on the liftability of group actions...

The Hunter–Saxton equation is the Euler equation for the geodesic flow on the quotient space of the infinite-dimensional group of orientation preserving diffeomorphisms of the unit circle modulo the subgroup of rigid rotations equipped with a right-invariant metric. We establish several properties of this quotient space: it has constant sectional curvature equal to 1, the Riemannian exponential...