Criteria of closedness of nilradicals in zero dimensional locally compact rings

We study in this paper conditions under which nilradicals of totally disconnected locally compact rings are closed. In the paper is given a characterization of locally finite compact rings via identities.

A new topology over the primary-like spectrum of a module

<p>Let R be a commutative ring with identity and M a unitary R-module. The primary-like spectrum Spec_L(M) is the collection of all primary-like submodules Q of M, the recent generalization of primary ideals, such that M/Q is a primeful R-module. In this article, we topologies Spec_L(M) with the patch-like topology, and show that when, Spec_L(M) with the patch-like topology is a quasi-compact, Hausdorff, totally disconnected space.</p>

Dendrogramic Representation of Data: CHSH Violation vs. Nonergodicity

This paper is devoted to the foundational problems of dendrogramic holographic theory (DH theory). We used the ontic–epistemic (implicate–explicate order) methodology. The epistemic counterpart is based on the representation of data by dendrograms constructed with hierarchic clustering algorithms. The ontic universe is described as a p-adic tree; it is zero-dimensional, totally disconnected, disordered, and bounded (in p-adic ultrametric spaces). Classical–quantum interrelations lose their sharpness; generally, simple dendrograms are “more quantum” than complex ones. We used the CHSH inequality as a measure of quantum-likeness. We demonstrate that it can be violated by classical experimental data represented by dendrograms. The seed of this violation is neither nonlocality nor a rejection of realism, but the nonergodicity of dendrogramic time series. Generally, the violation of ergodicity is one of the basic features of DH theory. The dendrogramic representation leads to the local realistic model that violates the CHSH inequality. We also considered DH theory for Minkowski geometry and monitored the dependence of CHSH violation and nonergodicity on geometry, as well as a Lorentz transformation of data.

Finitely generated subgroups and Chabauty topology in totally disconnected locally compact groups

For a locally compact group 𝐺, we write S ⁢ U ⁢ B ⁢ ( G ) {\mathcal{SUB}}(G) for the space of closed subgroups of 𝐺 endowed with the Chabauty topology. For any positive integer 𝑛, we associate to 𝐺 the function δ G , n \delta_{G,n} from G n G^{n} to S ⁢ U ⁢ B ⁢ ( G ) {\mathcal{SUB}}(G) defined by δ G , n ⁢ ( g 1 , … , g n ) = gp ¯ ⁢ ( g 1 , … , g n ) , \delta_{G,n}(g_{1},\ldots,g_{n})=\overline{\mathrm{gp}}(g_{1},\ldots,g_{n}), where gp ¯ ⁢ ( g 1 , … , g n ) \overline{\mathrm{gp}}(g_{1},\ldots,g_{n}) denotes the closed subgroup topologically generated by g 1 , … , g n g_{1},\ldots,g_{n} . It would be interesting to know for which groups 𝐺 the function δ G , n \delta_{G,n} is continuous for every 𝑛. Let [ HW ] [\mathtt{HW}] be the class of such groups. Some interesting properties of the class [ HW ] [\mathtt{HW}] are established. In particular, we prove that [ HW ] [\mathtt{HW}] is properly included in the class of totally disconnected locally compact groups. The class of totally disconnected locally compact locally pronilpotent groups is included in [ HW ] [\mathtt{HW}] . Also, we give an example of a solvable totally disconnected locally compact group not contained in [ HW ] [\mathtt{HW}] .

Dendrogramic Representation of Data: CHSH-Violation vs. Nonergodicity

This paper is devoted to the foundational problems of dendrogramic holographic theory (DH-theory). We use the ontic-epistemic (implicate-explicate order) methodology. The epistemic counterpart is based on representation of data by dendrograms constructed with hierarchic clustering algorithms. The ontic Universe is described as the p-adic tree; it is zero dimensional, totally disconnected, disordered, and bounded (in p-adic ultrametric). Interrelation classical-quantum loses its sharpness; generally simple dendrograms are ``more quantum&rsquo;&rsquo; than complex one. We use the CHSH-inequality as a measure of quantum(-likeness). We demonstrate that it can be violated by classical experimental data represented by dendrograms. The seed of this violation is neither nonlocality nor rejection of realism. This is nonergodicity of dendrogramic time series. Generally, violation of ergodicity is one of the basic features of DH-theory. We also consider DH-theory for Minkovski geometry and monitor the dependence of CHSH-violation and nonergodicity on geometry as well as a Lorentz transformation of data.

Fractals Parrondo’s Paradox in Alternated Superior Complex System

This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria.

Exotic Nonleaves with Infinitely Many Ends

We show that any simply connected topological closed $4$-manifold punctured along any compact, totally disconnected tame subset $\Lambda $ admits a continuum of smoothings, which are not diffeomorphic to any leaf of a $C^{1,0}$ codimension one foliation on a compact manifold. This includes the remarkable case of $S^4$ punctured along a tame Cantor set. This is the lowest reasonable regularity for this realization problem. These results come from a new criterion for nonleaves in $C^{1,0}$ regularity. We also include a new criterion for nonleaves in the $C^2$-category. Some of our smooth nonleaves are “exotic”, that is, homeomorphic but not diffeomorphic to leaves of codimension one foliations on a compact manifold, being the 1st examples in this class.

L 2-Betti Numbers of C*-Tensor Categories Associated with Totally Disconnected Groups

We prove that the $L^2$-Betti numbers of a rigid $C^*$-tensor category vanish in the presence of an almost-normal subcategory with vanishing $L^2$-Betti numbers, generalising a result of [ 7]. We apply this criterion to show that the categories constructed from totally disconnected groups in [ 6] have vanishing $L^2$-Betti numbers. Given an almost-normal inclusion of discrete groups $\Lambda &lt;\Gamma $, with $\Gamma $ acting on a type $\textrm{II}_1$ factor $P$ by outer automorphisms, we relate the cohomology theory of the quasi-regular inclusion $P\rtimes \Lambda \subset P\rtimes \Gamma $ to that of the Schlichting completion $G$ of $\Lambda &lt;\Gamma $. If $\Lambda &lt;\Gamma $ is unimodular, this correspondence allows us to prove that the $L^2$-Betti numbers of $P\rtimes \Lambda \subset P\rtimes \Gamma $ are equal to those of $G$.