We study the action of translation operators on wavelet subspaces. This action gives rise to an equivalence relation on the set of all wavelets. We show by explicit construction that each of the associated equivalence classes is non-empty.

In this paper we show how the defense relation among abstract arguments can be used to encode the reasons for accepting arguments. After introducing a novel notion of defenses and defense graphs, we propose a defense semantics together with a new notion of defense equivalence of argument graphs, and compare defense equivalence with standard equivalence and strong equivalence, respectively. Then...

Every equivalence relation can be made into a groupoid with the same underlying set if we define the multiplication as follows: xy = x if x, y are related; otherwise, xy = y. The groupoids, obtained in this way, are called equivalence algebras. We find a finite base for the equations of equivalence algebras. The base consists of equations in four variables, and we prove that there is no base co...

We formulate two new criteria of amenability of discrete equivalence relations: in terms of asymptotically invariant families of leafwise probability measures and in terms of isoperimetric properties of leafwise graph structures. These criteria lead to a geometric proof of the Connes{Feldman{Weiss theorem on coincidence of amenability and hyperrniteness for equivalence relations. Nous consid er...

Let T be a counterexample to Vaught’s conjecture. Let ET denote the isomorphism equivalence relation of T . Let Fω1 be the countable admissible ordinal equivalence relation defined on ω2 by x Fω1 y if and only if ωx 1 = ω y 1 . This note will show that in L and set-generic extensions of L, ¬(ET ≤∆11 Fω1 ). Using ZFC, if T is a non-minimal counterexample to Vaught’s conjecture, then ¬(ET ≤∆11 Fω...

The vague sum (product) of a finite number of real numbers and the vague integral multiples (powers) of real numbers based on a perfect vague addition (multiplication) operation are introduced, and some of their fundamental properties are pointed out. Furthermore, some examples are designed to show how they can be practically implemented. Keyword: Fuzzy arithmetic, Vague arithmetic, Vague arith...

We prove that the σ-ideal I(E) (of closed smooth sets with respect to a non-smooth Borel equivalence relation E) does not have the covering property. In fact, the same holds for any σ-ideal containing the closed transversals with respect to an equivalence relation generated by a countable group of homeomorphisms. As a consequence we show that I(E) does not have a Borel basis.

In this paper we investigate the dependence of recursively enumerable structures on the equality relation which is fixed to a specific r.e. equivalence relation. We compare r.e. equivalence relations on the natural numbers with respect to the amount of structures they permit to represent from a given class of structures such as algebras, permutations and linear orders. In particular, we show th...

We consider the harmonic measure on the Gromov boundary of a nonamenable hyperbolic group defined by a finite range random walk on the group, and study the corresponding orbit equivalence relation on the boundary. It is known to be always amenable and of type III. We determine its ratio set by showing that it is generated by certain values of the Martin kernel. In particular, we show that the e...

The notion of local subgroupoid as a generalisation of a local equivalence relation was defined in a previous paper by the first two authors. Here we the notion of star path connectivity for a Lie groupoid to give an important new class of examples, generalising the local equivalence relation of a foliation, and develop in this new context basic properties of coherence, due earlier to Rosenthal...