A defining feature of many large empirical networks is their intrinsic complexity. However, many networks also contain a large degree of structural repetition. An immediate question then arises: can we characterize essential network complexity while excluding structural redundancy? In this article we utilize inherent network symmetry to collapse all redundant information from a network, resulti...

There is a close relationship between Mumford’s geometric invariant theory (GIT) in (complex) algebraic geometry and the process of reduction in symplectic geometry. GIT was developed to construct quotients of algebraic varieties by reductive group actions and thus to construct and study moduli spaces [28, 29]. When a moduli space (or a compactification of a moduli space) over C can be construc...

We adapt direct, delayed, and fair simulation to alternating Büchi automata. Unlike with nondeterministic Büchi automata, naive quotients do not preserve the recognized language. As a remedy, we present specifically designed definitions of quotients, namely minimax and semi-elective quotients: minimax quotients, which are simple and have a minimum number of transitions, preserve the recognized ...

We generalize a major portion of the classical theory of Cand C embedded subspaces to pointfree topology, where the corresponding notions are frame Cand C -quotients. The central results characterize these quotients and generalize Urysohn’s Extension Theorem, among others. The proofs require calculations in CL, the archimedean f -ring of frame maps from the topology of the reals into the frame ...

We define and construct Ramanujan complexes. These are simplicial complexes which are higher dimensional analogues of Ramanujan graphs (constructed in [LPS]). They are obtained as quotients of the buildings of type Ad-1 associated with PGLd(F) where F is a local field of positive characteristic. 1. I n t r o d u c t i o n A finite k-regular graph X is called a Ramanujan graph if for every eigen...

In this article, a sequel to Global Frobenius Liftability I (math:1708:03777v2), we continue the development of comprehensive theory liftings modulo $p^2$. We study compatibility divisors and closed subschemes with liftings, blow-ups, descent under quotients by some group actions, stability base change, properties associated F-splittings. Consequently, characterise liftable surfaces Fano threef...

In this paper we show that every ideal with linear quotients is componentwise linear. We also generalize the Eliahou-Kervaire formula for graded Betti numbers of stable ideals to homogeneous ideals with linear quotients.

In this paper, we first define the notion of finitely-cosmall quotient (singly-cosmall quotient) morphisms. Then give a characterization new concept. We show that an epimorphism p:Y→U is if and only for any right R-module Z morphism g:Z→Y such pg finitely-copartial isomorphism (singly-copartial isomorphism) from to Y with codomain U finitely (singly) split epimorphism. also investigate relation...