Throughout this paper, we will establish a comprehensive theoretical foundation and rigorously develop the methodology to investigate the structure of quotient ring S/P under the influence of symmetric generalized n-derivations, where S represents an arbitrary ring, and P signifies a prime ideal of S satisfying certain algebraic identities acting on prime ideal P without relying on the assumption of primeness or semi-primeness of the ring.

A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation is central, then the ring is commutative. In the present paper, we extend this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we establish *-version of Vukman’s theorem [48, Theorem 1]. Precisely, we describe the structure of quotient ring A/L, where A is an arbitrary ring and L is a prime ideal of A. Further, by taking advantage of the *-version of Vukman’s theorem, we show that if a 2-trosion free semiprime ring A with involution admits a nonzero *-centralizing derivation, then A contains a nonzero central ideal. This result is in a spirit of the classical result due to Bell and Martindale [19, Theorem 3]. As the applications, we extends and unify several classical theorems proved in [6],[25,[42], and [48] . Finally, we conclude our paper with a direction for further research.

Let $m$ and $n$ be the fixed positive integers. Suppose $\mathcal{A}$ is a von Neuman algebra with no central summands of type $I_{1}$ and $L_{m}$ be a Lie type higher derivation i.e., an additive (linear) map $L_{m} :\mathcal{A}\to \mathcal{A}$ such that \begin{equation}\label{def2} \[L_{m}(p_{n}(\mathfrak{S}_{1},\mathfrak{S}_{2},\cdots,\mathfrak{S}_{n}))=\sum_{l_1+l_2+\cdots+l_n=m}p_{n}\big(L_{l_1}(\mathfrak{S}_{1}),L_{l_2}(\mathfrak{S}_{2}),\cdots,L_{l_n}(\mathfrak{S}_{n})\big)\] %L_{m}(p_{n}(\mathfrak{S}_{1},\mathfrak{S}_{2},\cdots,\mathfrak{S}_{n}))=\sum_{l_1+l_2+\cdots+l_n=m}p_{n}\big(L_{l_1}(\mathfrak{S}_{1}),L_{l_2}(\mathfrak{S}_{2}),\cdots,L_{l_n}(\mathfrak{S}_{n})\big)\nonumber \end{equation} for all $\mathfrak{S}_{1},\mathfrak{S}_{2},\cdots,\mathfrak{S}_{n}\in \mathcal{A}$. In the present paper, we study Lie type higher derivations on von Neuman algebras and prove that every additive Lie type higher derivation on $\mathcal{A}$ has a standard form at zero product as well as at projection product. Further, we discuss some more related results.

Let $n \geq 2$ be a fixed integer and $\mathcal{A}$ be a $C^*$-algebra. A permuting $ n $-linear map $ \mathcal{G} : \mathcal{A} ^{n} \rightarrow \mathcal{A} $ is known to be symmetric generalized $n$-derivation if there exists a symmetric $n$-derivation $ \mathfrak{D} : \mathcal{A} ^{n} \rightarrow \mathcal{A} $ such that $ \mathcal{G} \left(x_{1}, x_{2}, \ldots, x_{i} x_{i}^{\prime}, \ldots, x_{n}\right)= \mathcal{G} \left(x_{1}, x_{2}, \ldots, x_{i}, \ldots, x_{n}\right) x_{i}^{\prime}+x_{i} \mathfrak{D} (x_{1}, x_{2},\ldots, x_{i}^{\prime}, \ldots, x_{n})$ holds for all $x_{i}, x_{i}^{\prime} \in \mathcal{A} $. In this paper, we investigate the structure of $C^*$-algebras involving generalized linear $n$-derivations. Moreover, we describe the forms of traces of linear $n$-derivations satisfying certain functional identity.

This paper proposes a curved link-slab (CLS) structure, simplified into a hingeless-arch model, to address the current cracking phenomenon of CLS concrete. The stress formula of the hingeless arch under various loads is derived based on the classical mechanic's method. Based on an actual bridge example, the mechanical properties of CLS are analyzed under different loads and load combinations. The results show that: (1) the CLS stress is significantly lower than that of the flat link-slab structure (FLS), (2) its stress values are less than the concrete tensile limit, and (3) the CLS can effectively solve the concrete cracking phenomenon on the link-slab. The rationality of the stress formula derived from the simplified model of the hingeless arch is verified using the finite element method (FEM). The parametric sensitivity analysis shows that variation of the reinforcement ratio of the CLS has a limited impact on it. Considering both the concrete tensile and compressive limit, the thickness of the CLS should be 15 cm to 20 cm, and its design span should be about 5% to7.5% of the main beam length.

Lie algebra plays an important role in the study of singularity theory and other field of sciences. Finding numerous invariants linked with isolated singularities is always a main interest in the classification theory of isolated singularities. Any Lie algebra that characterizes simple singularity produces a natural question. The study of properties such as to find the dimensions of newly defined algebra is a remarkable work. Hussain, Yau and Zuo \cite{HYZ10} have been found a new class of Lie algebra \texorpdfstring{ $ \mathcal{L}_ k (V)$}{LG}, \texorpdfstring{$k\geq 1$}{LG} i.e., Der ($M_{k}(V), M_{k}(V)$) and purposed a conjecture over its dimension $\delta_{k}(V)$ for $k\geq0$. Later they proved it true for $k$ up to $k=1,2,3,4,5$. In this work, the main concern is whether it's true for a higher value of $k$. According to this, we calculate first, the dimension of Lie algebra \texorpdfstring{$\mathcal{L}_ k (V)$}{LG} for $k=6$ and then compute the upper estimate conjecture of fewnomial isolated singularities. Along with, we also justify the inequality conjecture: \texorpdfstring{$\delta_{k+1}(V) < \delta_{k}(V)$}{LG} for $k=6$. Our calculated results are innovative and a new addition to the study of singularity theory.

This paper presents an improved droop control strategy for grid-connected inverter power stability and power quality under distorted with consideration of grid fluctuation and inter-harmonics. An instantaneous frequency without PLL and amplitude of capacitor voltage feed-forward control strategy is given to power stability control, meanwhile a grid current feedback control is given by an incomplete derivation with a high-pass filter, so that the harmonics and inter-harmonics current can be suppressed. These approaches can provide both good active and reactive power dynamic response under fluctuation of frequency, and rejection ability against harmonic and inter-harmonic voltage. Based on model of inverter, the proposed control strategies are designed in detail. Simulations and experiments are present to validate the effectiveness of proposed method.

There are rich proper noun derivative semantic relationships in geographic entity vector data. The identification of traditional proper noun derivative semantic relationships usually requires toponym researchers to examine the source of derived toponyms by consulting a large number of toponym documents. This method cannot adapt to the mining of proper noun derivative semantic relationships for large-scale geographical entity. In this regard, this paper proposes a method of identifying the proper noun derivative semantic relationships for geographical entity by combining prompt learning and XGBoost model. This method transforms the extraction problem of proper noun derivative semantic relationships into a binary classification problem by analyzing the characteristics of the derivation relationship of toponyms in the form of toponyms and the spatial and temporal distribution of geography. Firstly, prompt learning and gaussian mixture model are used to extract and cluster the derivation words of toponyms, so as to realize the recognition of derived toponyms collections. Then, enumerate the possible proper noun derivations in the toponym set, and then the XGBoost model is used to identify the semantic relations derived from the proper noun of geographical entities. In the experiment, the recognition method of proper noun derivative semantic relationships based on prompt learning and XGBoost proposed in this paper has achieved good relationship recognition effect in the experiment. This method has certain practical value in the fields of toponym management and toponym translation.

Abstract:Shown is the derivation of Lorentz-Einstein k-factor in SRT as an amplitude-term of oscillation-differential equations of second order.This case is shown for classical Lorentz-factor as solution of an equation for undamped oscillation as well as the developed theorem as a second solution for advanced SRT of fourth order with an equation for damped oscillation-states.This advanced term allows a calculation for any velocities by real rest mass