Ideal-based zero-divisor graph of MV-algebras

Let $(A, \oplus, *, 0)$ be an MV-algebra, $(A, \odot, 0)$ be the associated commutative semigroup, and $I$ be an ideal of $A$. Define the ideal-based zero-divisor graph $\Gamma_{I}(A)$ of $A$ with respect to $I$ to be a simple graph with the set of vertices $V(\Gamma_{I}(A))=\{x\in A\backslash I ~|~ (\exists~ y\in A\backslash I) ~x\odot y\in I\},$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if $x\odot y\in I$. We prove that $\Gamma_{I}(A)$ is connected and its diameter is less than or equal to $3$. Also, some relationship between the diameter (the girth) of $\Gamma_{I}(A)$ and the diameter (the girth) of the zero-divisor graph of $A/I$ are investigated. And using the girth of zero-divisor graphs (resp. ideal-based zero-divisor graphs) of MV-algebras, we classify all MV-algebras into $2~($resp. $3)$ types

A General Synthesis Strategy for Hierarchical Porous Metal Oxide Hollow Spheres

The hierarchical porous TiO2 hollow spheres were successfully prepared by using the hydrothermally synthesized colloidal carbon spheres as templates and tetrabutyl titanate as inorganic precursors. The diameter and wall thickness of hollow TiO2 spheres were determined by the hard templates and concentration of tetrabutyl titanate. The particle size, dispersity, homogeneity, and surface state of the carbon spheres can be easily controlled by adjusting the hydrothermal conditions and adding certain amount of the surfactants. The prepared hollow spheres possessed the perfect spherical shape, monodispersity, and hierarchically pore structures, and the further experiment verified that the present approach can be used to prepare other metal oxide hollow spheres, which could be used as catalysis, fuel cells, lithium-air battery, gas sensor, and so on

Language-Specific Representation of Emotion-Concept Knowledge Causally Supports Emotion Inference

Understanding how language supports emotion inference remains a topic of debate in emotion science. The present study investigated whether language-derived emotion-concept knowledge would causally support emotion inference by manipulating the language-specific knowledge representations in large language models. Using the prompt technique, 14 attributes of emotion concepts were found to be represented by distinct artificial neuron populations. By manipulating these attribute-related neurons, the majority of the emotion inference tasks showed performance deterioration compared to random manipulations. The attribute-specific performance deterioration was related to the importance of different attributes in human mental space. Our findings provide causal evidence in support of a language-based mechanism for emotion inference and highlight the contributions of emotion-concept knowledge.Comment: 39 pages, 13 figures, 2 tables, fix formatting error

Detection of neural connections with ex vivo MRI using a ferritin-encoding trans-synaptic virus

The elucidation of neural networks is essential to understanding the mechanisms of brain functions and brain disorders. Neurotropic virus-based trans-synaptic tracing tools have become an effective method for dissecting the structure and analyzing the function of neural-circuitry. However, these tracing systems rely on fluorescent signals, making it hard to visualize the panorama of the labeled networks in mammalian brain in vivo. One MRI method, Diffusion Tensor Imaging (DTI), is capable of imaging the networks of the whole brain in live animals but without information of anatomical connections through synapses. In this report, a chimeric gene coding for ferritin and enhanced green fluorescent protein (EGFP) was integrated into Vesicular stomatitis virus (VSV), a neurotropic virus that is able to spread anterogradely in synaptically connected networks. After the animal was injected with the recombinant VSV (rVSV), rVSV-Ferritin-EGFP, into the somatosensory cortex (SC) for four days, the labeled neural-network was visualized in the postmortem whole brain with a T2-weighted MRI sequence. The modified virus transmitted from SC to synaptically connected downstream regions. The results demonstrate that rVSV-Ferritin-EGFP could be used as a bimodal imaging vector for detecting synaptically connected neural-network with both ex vivo MRI and fluorescent imaging. The strategy in the current study has the potential to longitudinally monitor the global structure of a given neural-network in living animals

A study of unit graphs and unitary cayley graphs associated with rings

In this thesis, we study the unit graph G(R) and the unitary Cayley graph Γ(R) of a ring R, and relate them to the structure of the ring R. Chapter 1 gives a brief history and background of the study of the unit graphs and unitary Cayley graphs of rings. Moreover, some basic concepts, which are needed in this thesis, in ring theory and graph theory are introduced. Chapter 2 concerns the unit graph G(R) of a ring R. In Section 2.2, we first prove that the girth gr(G(R)) of the unit graph of an arbitrary ring R is 3, 4, 6 or ∞. Then we determine the rings R with R/J(R) semipotent and with gr(G(R)) = 6 or ∞, and classify the rings R with R/J(R) right self-injective and with gr(G(R)) = 3 or 4. The girth of the unit graphs of some ring extensions are also investigated. The focus of Section 2.3 is on the diameter of unit graphs of rings. We prove that diam(G(R)) ∈ {1, 2, 3,∞} for a ring R with R/J(R) self-injective and determine those rings R with diam(G(R)) = 1, 2, 3 or ∞, respectively. It is shown that, for each n ≥ 1, there exists a ring R such that n ≤ diam(G(R)) ≤ 2n. The planarity of unit graphs of rings is discussed in Section 2.4. We completely determine the rings whose unit graphs are planar. In the last section of this chapter, we classify all finite commutative rings whose unit graphs have genus 1, 2 and 3, respectively. Chapter 3 is about the unitary Cayley graph Γ(R) of a ring R. In Section 3.2, it is proved that gr(Γ(R)) ∈ {3, 4, 6,∞} for an arbitrary ring R, and that, for each n ≥ 1, there exists a ring R with diam(Γ(R)) = n. Rings R with R/J(R) self-injective are classified according to diameters of their unitary Cayley graphs. In Section 3.3, we completely characterize the rings whose unitary Cayley graphs are planar. In Section 3.4, we prove that, for each g ≥ 1, there are at most finitely many finite commutative rings R with genus γ(Γ(R)) = g. We also determine all finite commutative rings R with γ(Γ(R)) = 1, 2, 3, respectively. Chapter 4 is about the isomorphism problem between G(R) and Γ(R). We prove that for a finite ring R, G(R) ∼= Γ(R) if and only if either char(R/J(R)) = 2 or R/J(R) = Z2 × S for some ring S

On the binary system of factors of formal matrix rings

summary:We investigate the formal matrix ring over $R$ defined by a certain system of factors. We give a method for constructing formal matrix rings from non-negative integer matrices. We also show that the principal factor matrix of a binary system of factors determine the structure of the system