Initial segments of computable linear orders with computable natural relations

We study the algorithmic complexity of natural relations on initial segments of computable linear orders. We prove that there exists a computable linear order with computable density relation such that its Π₁⁰-initial segment has no computable presentation with a computable density relation. We also prove that the same holds for a right limit and a left limit relations.     

Means on equivalence relations

Suppose that X is a Polish space and E is a countable Borel equivalence relation on X. We show that if there is a Borel assignment of means to the equivalence classes of E, then E is smooth. We also show that if there is a Baire measurable assignment of means to the equivalence classes of E, then E is generically smooth. Research partially supported by NSF Grants DMS-9987437 and DMS-0455285. Research partially supported by NSF VIGRE Grant DMS-0502315.     

Sofic equivalence relations

We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes' Embedding Conjecture as well as the Measurable Determinant Conjecture of Lück, Sauer and Wegner hold for treeable equivalence relations.     

Local equivalence relations

In this paper, we investigate the concept of local equivalence relation, a notion suggested by Grothendieck. A local equivalence relation on a topological space X is a global section of the sheaf of germs of equivalence relations on X. We investigate the extent to which a local equivalence relation can be described by a global one and analogously when can a global equivalence relation be recovered from its associated local one. We also look at the notion of a fiber map, which sheds further light on these concepts. A motivating example is that of a foliation on a manifold.     

Quasi-proper meromorphic equivalence relations

The aim of this article is to complete and generalize results of (Mathieu Ann. Inst. Fourier (Grenoble) t. 50:1155–1189, 2000) and (Barlet Comment. Math. Helv. 83:869–888, 2008) and to show that they imply a rather general existence theorem for meromorphic quotients of strongly quasi-proper meromorphic equivalence relations. In this context, generic equivalence classes are asked to be pure dimensional closed analytic subset with finitely many irreducible components. As an application of these methods we prove a Stein factorization theorem for a strongly quasi-proper map.     

Homological equivalence relations on an algebraic curve

We study the collection of homological equivalence relations on a fixed curve. We construct a moduli space for pairs consisting of a curve of genus g and a homological equivalence relation of degree n on the curve, and a classifying set for homological equivalence relations of degree n on a fixed curve, modulo automorphisms of the curve. We identify a special type of homological equivalence relations, and we characterize the special homological equivalence relations in terms of the existence of elliptic curves in the Jacobian of the curve.     

Relative exchangeability with equivalence relations

We describe an Aldous–Hoover-type characterization of random relational structures that are exchangeable relative to a fixed structure which may have various equivalence relations. Our main theorem gives the common generalization of the results on relative exchangeability due to Ackerman (Representations of \(\text {Aut}(\mathcal {M})\)-invariant measures: part I, 2015. arXiv:1509.06170) and Crane and Towsner (Relatively exchangeable structures, 2015) and hierarchical exchangeability results due to Austin and Panchenko (Probab Theory Relat Fields 159(3–4):809–823, 2014).     

Endotypes of partial equivalence relations

We classify all partial equivalence relations according to their endotype with respect to endotopisms and compute the cardinalities of endotopism semigroups (strong endotopism monoids, autotopism groups) of partial equivalence relations on a finite set.

     

Indecomposability of treed equivalence relations

We define rigorously a “treed” equivalence relation, which, intuitively, is an equivalence relation together with a measurably varying tree structure on each equivalence class. We show, in the nonamenable, ergodic, measure-preserving case, that a treed equivalence relation cannot be stably isomorphic to a direct product of two ergodic equivalence relations.     

Fuzzy probability measures

In [4] Höhle has defined fuzzy measures on G-fuzzy sets [2] where G stands for a regular Boolean algebra. Consequently, since the unit interval is not complemented, fuzzy sets in the sense of Zadeh [8] do not fit in this framework in a straightforward manner. It is the purpose of this paper to continue the work started in [5] which deals with [0,1]-fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space [5]. We give necessary and sufficient conditions for such a measure to be a classical integral as in [9] in the case the space is generated. A counterexample in the general case is also presented. Finally it is shown that a fuzzy probability measure is always an integral (if the space is generated) if we replace the operations ∧ and ∨ by the t-norm T_o and its dual S₀ (see [6]).     

Banach-valued probability measures

We introduce and study the notion of Banach-valued probability measures on a compact semitopological semigroup. In particular, we prove that these measures are nontrivial idempotents and convolution is separately continuous. We give an example of a Banach-valued measure where the support may not be simple; though for any idempotent measure the support is a closed simple subsemigroup.     

Determinantal probability measures

Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We study basic combinatorial and probabilistic aspects in the discrete case. Our main results concern relationships with matroids, stochastic domination, negative association, completeness for infinite matroids, tail triviality, and a method for extension of results from orthogonal projections to positive contractions. We also present several new avenues for further investigation, involving Hilbert spaces, combinatorics, homology, and group representations, among other areas.