Given a Polish groupG, let$E(G)$be the right coset equivalence relation$G^{\omega }/c(G)$, where$c(G)$is the group of all convergent sequences inG. The connected component of the identity of a Polish groupGis denoted by$G_0$.Let$G,H$be locally compact abelian Polish groups. If$E(G)\leq _B E(H)$, then there is a continuous homomorphism$S:G_0\rightarrow H_0$such that$\ker (S)$is non-archimedean. The converse is also true whenGis connected and compact.For$n\in {\mathbb {N}}^+$, the partially ordered set$P(\omega )/\mbox {Fin}$can be embedded into Borel equivalence relations between$E({\mathbb {R}}^n)$and$E({\mathbb {T}}^n)$.

We investigate Borel reducibility between equivalence relations $E(X;p)=X^{\mathbb{N}}/\ell_{p}(X)'s$ where X is a separable Banach space. We show that this reducibility is related to the so called Hölder(α) embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between E(L r ; p)'s and E(c 0 ; p)'s for r, p ∈ [1, +∞). We also answer a problem presented by Kanovei in the affirmative by showing that $\mathrm{C}\left({\mathrm{\mathbb{R}}}^{+}\right)/{\mathrm{C}}_{0}\left({\mathrm{\mathbb{R}}}^{+}\ri ght)$ (...) is Borel bireducible to $\mathbb{R}^{\mathbb{N}}/c_{0}$. (shrink)

Let , be a sequence of pseudo-metric spaces, and let p≥1. For , let . For Borel reducibility between equivalence relations , we show it is closely related to finitely Hölder embeddability between pseudo-metric spaces.

In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by $\alpha $ -CLI and L- $\alpha $ -CLI where $\alpha $ is a countable ordinal. We establish three results: (1) G is $0$ -CLI iff $G=\{1_G\}$ ; (2) G is $1$ -CLI iff G admits a compatible complete two-sided invariant metric; and (3) G is L- $\alpha $ -CLI iff G is locally $\alpha $ (...) -CLI, i.e., G contains an open subgroup that is $\alpha $ -CLI. Subsequently, we show this hierarchy is proper by constructing non-archimedean CLI Polish groups $G_\alpha $ and $H_\alpha $ for $\alpha <\omega _1$, such that: (1) $H_\alpha $ is $\alpha $ -CLI but not L- $\beta $ -CLI for $\beta <\alpha $ ; and (2) $G_\alpha $ is $(\alpha +1)$ -CLI but not L- $\alpha $ -CLI. (shrink)

Memristor is the fourth basic electronic element discovered in addition to resistor, capacitor, and inductor. It is a nonlinear gadget with memory features which can be used for realizing chaotic, memory, neural network, and other similar circuits and systems. In this paper, a novel memristor-based fractional-order chaotic system is presented, and this chaotic system is taken as an example to analyze its dynamic characteristics. First, we used Adomian algorithm to solve the proposed fractional-order chaotic system and yield a chaotic phase (...) diagram. Then, we examined the Lyapunov exponent spectrum, bifurcation, SE complexity, and basin of attraction of this system. We used the resulting Lyapunov exponent to describe the state of the basin of attraction of this fractional-order chaotic system. As the local minimum point of Lyapunov exponential function is the stable point in phase space, when this stable point in phase space comes into the lowest region of the basin of attraction, the solution of the chaotic system is yielded. In the analysis, we yielded the solution of the system equation with the same method used to solve the local minimum of Lyapunov exponential function. Our system analysis also revealed the multistability of this system. (shrink)

In this paper, notions of ‐like and c0‐like equivalence relations are introduced. We characterize the positions of ‐like and c0‐like equivalence relations in the Borel reducibility hierarchy by comparing them with equivalence relations and.

We prove the Decomposability Conjecture for functions of Baire class $2$ from a Polish space to a separable metrizable space. This partially answers an important open problem in descriptive set theory.

Let X, Y be Polish spaces,,. We say A is universal for Γ provided that each x‐section of A is in Γ and each element of Γ occurs as an x‐section of A. An equivalence relation generated by a set is denoted by, where. The following results are shown: If A is a set universal for all nonempty closed subsets of Y, then is a equivalence relation and. If A is a set universal for all countable subsets of Y, then (...) is a equivalence relation, and and ; if, then ; if every set is Lebesgue measurable or has the Baire property, then. for, if every set has the Baire property, and E is any equivalence relation, then. (shrink)

We investigate diagonal actions of Polish groups and the related intersection operator on closed subgroups of the acting group. The Borelness of the diagonal orbit equivalence relation is characterized and is shown to be connected with the Borelness of the intersection operator. We also consider relatively tame Polish groups and give a characterization of them in the class of countable products of countable abelian groups. Finally an example of a logic action is considered and its complexity in the Borel reducbility (...) hierarchy determined. (shrink)