Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of a finitely generated algebra of invariants. As an application, w...

We consider the analytic continuation of $(p+q)$-dimensional Minkowski space (with $p$ and $q$ even) to $(p,q)$-signature, study conformal boundary resulting "Klein space". Unlike familiar $(-+++..)$ signature, now null infinity ${\mathcal I}$ has only one connected component. The spatial timelike infinities ($i^0$ $i'$) are quotients generalizations AdS spaces non-standard signature. Together,...

The past research on the state complexity of operations on regular languages is examined, and a new approach based on an old method (derivatives of regular expressions) is presented. Since state complexity is a property of a language, it is appropriate to define it in formal-language terms as the number of distinct quotients of the language, and to call it “quotient complexity”. The problem of ...

The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rectangular Rayleigh Quotients. Many useful properties of eigenvalues stem are from the Courant-Fischer minimax theorem, from Weyl’s theorem, and their corollaries. Another aspect rega...

The moduli spaceM0,n plays important roles in algebraic geometry and theoretical physics. Yet, some basic properties of M 0,n still remain open. For example, M0,n is rational and nearly toric (that is, it contains a toric variety as a Zariski open subset), but it is not a toric variety itself starting from dimension 2 (n ≥ 5). So, a basic question is: Can it be degenerated flatly to a projectiv...

The moduli space of stable quotients introduced by Marian-Oprea-Pandharipande provides a natural compactification of the space of morphisms from nonsingular curves to a nonsingular projective variety and carries a natural virtual class. We show that the analogue of Givental’s J-function for the resulting twisted projective invariants is described by the same mirror hypergeometric series as the ...

Every finite group acts as the full isometry group of some compact hyperbolic 3-manifold. In this paper we study those finite groups which act maximally, that is when the ratio |Isom(M)|/vol(M) is maximal among all such manifolds. In two dimensions maximal symmetry groups are called Hurwitz groups, and arise as quotients of the (2,3,7)–triangle group. Here we study quotients of the minimal co-v...

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck’s Quot scheme. Over nodal curves, a relative construction is made to keep the torsion of the quotient away from the singularities. New compactifications of classical spaces arise naturally: a nonsingular and irreducible compactification of the ...

Inspired by Riemann’s work on certain quotients of the Dedekind Eta function, in this paper we investigate the value distribution of quotients of values of the Dedekind Eta function in the complex plane, using the form η(Ajz) η(Aj−1z) , where Aj−1 and Aj are matrices whose rows are the coordinates of consecutive visible lattice points in a dilation XΩ of a fixed region Ω in R, and z is a fixed ...

This article is motivated by the following local-to-global question: is every variety with tame quotient singularities globally the quotient of a smooth variety by a finite group? We show that this question has a positive answer for all quasi-projective varieties which are expressible as a quotient of a smooth variety by a split torus (e.g. simplicial toric varieties). Although simplicial toric...