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The entropy power inequality (EPI) and the Brascamp-Lieb inequality (BLI) are fundamental inequalities concerning the differential entropies of linear transformations of random vectors. The EPI provides lower bounds for the differential entropy of linear transformations of random vectors with independent components. The BLI, on the other hand, provides upper bounds on the differential entropy of a random vector in terms of the differential entropies of some of its linear transformations. In this paper, we define a family of entropy functionals, which we show are subadditive. We then establish that Gaussians are extremal for these functionals by mimicking the idea in Geng and Nair (2014). As a consequence, we obtain a new entropy inequality that generalizes both the BLI and EPI. By considering a variety of independence relations among the components of the random vectors appearing in these functionals, we also obtain families of inequalities that lie between the EPI and the BLI., Comment: 38 pages, 1 figure. Submitted to the IEEE Transactions on Information Theory for possible publication

The stability analysis problem is considered for multiarea load frequency control (LFC) systems with electric vehicles (EVs) and time delays. The novel linear operator inequality approach is proposed and a less conservative delay-dependent stability condition is achieved. First, the model of the multiarea LFC system with EVs and delays is expressed by the partial integral equation (PIE) at the first time. Then, a complete quadratic Lyapunov-Krasovskii functional is built in the form of an inner product of the linear partial integral (PI) operator. The novel stability criteria with less conservatism are proposed in the form of linear operator inequality. Moreover, the relationships between the delay margins and the controller parameters are shown. Finally, the simulations are conducted on both one-area and two-area LFC systems to show the effectiveness of the proposed approach.

A finite-time observer for a class of linear time-varying systems with bounded unknown inputs is presented in this note. The design of the observer exploits structural properties of the system and, through a linear operator, allows applying the robust-exact differentiator in a block form without requiring additional stabilizing terms. The proposed observer provides an exact estimate of the states after a finite transient-time despite the possible instability of the system and the effects of bounded unknown inputs.

Linear transformations are the dominating computation within many important applications. The natural multiply and accumulate feature of memristor crossbar arrays promise unprecedented processing capabilities to resistive dot-product engines (DPEs), which can accelerate approximate matrix-vector multiplication (MVM). Unfortunately, the precision of the analog computation may be degraded by parasitics, non-linear device characteristics, and variations. In this paper, we propose a framework called XMAP for mapping an arbitrary matrix into appropriate memristor conductance values (or state variables for non-linear devices). The specified conductance values are next programmed to the memristor hardware using accurate closed-loop tuning. XMAP is based on formulating the mapping problem as a mathematical optimization problem, which can be elegantly minimized using the concept of representable matrices, i.e., the matrices that can be represented on a crossbar. Compared with the state-of-the-art conversion algorithm, the computational accuracy is improved with up to 3.29X at the expense of overhead in run-time. The precision improvements translate into noteworthy application level benefits within signal compression and neural network inference.

The high-precision parameters in distribution networks are dif-ficult to obtain, which brings difficulties to the model-based methods and analysis. With the widespread deployment of high-precision measurement units, data-driven methods have greater advantages in practice. In addition, with massive integra-tion of distributed renewable generation and fast electric vehicle chargers, the fluctuations of net load increase significantly. The data-driven power flow calculation method based on the linear model becomes difficult to obtain accurate results for the low nonlinear adaptability. To improve the data-driven power flow calculation accuracy under high penetration of renewable distri-bution generation, this paper proposed an approach with high adaptability to the nonlinearity of power flow. Based on the thought of Koopman operator theory, the nonlinear relationship in power flow calculation is converted into a linear mapping in a higher dimension state space, which can significantly improve the calculation accuracy. Case studies on different IEEE cases have demonstrated that the proposed method can realize higher accu-racy in power flow calculation with large-scale power fluctuations, compared to the existing data-driven method, in both mesh and radial networks. Finally, measurement data of a practical 10kV distribution network has been further used to verify the effec-tiveness of the proposed method in practical applications.

The Kullback-Leibler divergence (KLD), which is widely used to measure the similarity between two distributions, plays an important role in many applications. In this article, we address the KLD metric-learning task, which aims at learning the best KLD-type metric from the distributions of datasets. Concretely, first, we extend the conventional KLD by introducing a linear mapping and obtain the best KLD to well express the similarity of data distributions by optimizing such a linear mapping. It improves the expressivity of data distribution, which means it makes the distributions in the same class close and those in different classes far away. Then, the KLD metric learning is modeled by a minimization problem on the manifold of all positive-definite matrices. To deal with this optimization task, we develop an intrinsic steepest descent method, which preserves the manifold structure of the metric in the iteration. Finally, we apply the proposed method along with ten popular metric-learning approaches on the tasks of 3-D object classification and document classification. The experimental results illustrate that our proposed method outperforms all other methods.

This paper presents a closed-loop time-varying continuous-time recursive subspace-based prediction method utilizing principle angles rotation. A simple linear mapping can be provided by generalized Poisson moment functionals, which can deal with the time-derivatives problems of input–output Hankel matrices. The parity space employed in fault detection field is adopted instead of using the observable subspace. The system matrices are estimated consistently by the instrumental variable method and principal component analysis, which solves the identification problems of biased results for the system operating in closed-loop with a feedback controller. The system matrices are predicted by the principle angles rotation of the signal subspaces spanned from the extended observability matrices. The effectiveness of the proposed method is illustrated by the numerical simulations and real applications.

In the paper (Hill, 1973) from 1973 R.D. Hill studied linear matrix maps L : ℂ q × q → ℂ n × n which map Hermitian matrices to Hermitian matrices, or equivalently, preserve adjoints, i.e., L ( V ∗ ) = L ( V ) ∗ , via representations of the form L ( V ) = ∑ k , l = 1 m H k l A l V A k ∗ , V ∈ ℂ q × q , for matrices A 1 , … , A m ∈ ℂ n × q and continued his study of such representations in later work, sometimes with co-authors, to completely positive matrix maps and associated matrix reorderings. In this paper we expand the study of such representations, referred to as Hill representations here, in various directions. In particular, we describe which matrices A 1 , … , A m can appear in Hill representations (provided the number m is minimal) and determine the associated Hill matrix H = H k l explicitly. Also, we describe how different Hill representations of L (again with m minimal) are related and investigate further the implication of ∗ -linearity on the linear map L .

Weyl's unitary matrices, which were introduced in Weyl's 1927 paper [12] on group theory and quantum mechanics, are p × p unitary matrices given by the diagonal matrix whose entries are the p-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by u and v , satisfy u p = v p = 1 p (the p × p identity matrix) and the commutation relation u v = ζ v u , where ζ is a primitive p-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if u and v are any d × d unitary matrices such that u p = v p = 1 d and u v = ζ v u , then there exists a unital completely positive linear map ϕ : M p ( C ) → M d ( C ) such that ϕ ( u ) = u and ϕ ( v ) = v . We also show, moreover, that any two pairs of p-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent, but that the assertion for three such unitaries fails. There is a standard tensor-product construction involving the Pauli matrices that produces irreducible sequences of anticommuting selfadjoint unitary matrices of arbitrary length. The matrices in this sequence are called Weyl-Brauer unitary matrices [11, Definition 6.63] . This standard construction is generalised herein to the case p ≥ 3 , producing a sequence of matrices that we also call Weyl-Brauer unitary matrices. We show that the Weyl-Brauer unitary matrices, as a g-tuple, are extremal in their matrix range, using recent ideas from noncommutative convexity theory.

Let U be any of the two exceptional Cartan factors V and VI and let δ : U → U be a Jordan *-triple derivation of U, that is, a map (neither linearity nor continuity of δ is assumed) that satisfies the functional equation δ { a b ⁎ a } = { δ ( a ) b ⁎ a } + { a δ ( b ) ⁎ a } + { a b ⁎ δ ( a ) } , ( a , b ∈ U ) , where ( a , b ) ↦ { a b ⁎ a } stands for the Jordan triple product in U. We give an explicit representation of δ as certain multipliers on U and prove that δ automatically is a continuous real linear map on U. This gives a new description of the real Banach Lie algebra of Jordan triple derivations of U.

Relative radiometric normalization (RRN) is widely used for radiometric calibration of bitemporal multispectral images prior to any temporal analysis such as change detection. However, standard RRN methods are not robust against anomalous (or changed) pixels, which warp the calibration and decrease the spectral similarity of processed images. This letter proposes a novel random sample consensus-based RRN method, which only uses small pixel subsets to implement the linear mapping relationship for RRN, and does not require the calibration of its parameters. The experimental results show that the proposed method performs favorably against the widely used RRN methods in all metrics considered in this letter.

The end-to-end scalable cascaded convolutional neural networks (SC-CNNs) are proposed to solve inverse scattering problems (ISPs), and the high-resolution image can be directly obtained from the scattered field with the guiding by multiresolution labels in the cascaded blocks. To alleviate the difficulty of solving the ISPs via a full-wave way, the proposed SC-CNNs are physically decomposed into two parts, i.e., the linear transformation and the multiresolution imaging networks. The first part is composed of one CNN block and is used to mimic the linear transformation [e.g., backpropagation (BP)] from scattered field to the preliminary image, whereas the second part consists of a few cascaded CNN blocks to realize the reconstruction from the rough image to high-resolution image. With more high-frequency components incorporating into the multiresolution labels, the cascaded networks can be guided through those labels, avoiding black-box operations and enhancing the physical meaning and interpretability. The proposed SC-CNNs are verified by both the synthetic and experimental examples and it is proved that better performance can be achieved in terms of both inversion accuracy and efficiency compared to the BP-Unet and direct inversion scheme (DIS).

Prestack amplitude variation with offset (AVO) inversion can provide abundant reservoir information underground, which is always implemented trace-by-trace. However, it cannot guarantee the lateral accuracy of the inversion results. To utilize the lateral difference of the angle gathers and improve the lateral resolution, the difference angle gathers are introduced. Based on the Bayes inversion framework, the objective function considering the difference angle gathers is first constructed. Then, the effect of difference angle gathers on inversion results is analyzed, which is essential to improve the accuracy of the inversion results. To further figure out the applicable conditions of difference angle gathers, different forward operators are analyzed including a nonlinear operator and a linear operator. The used nonlinear operator is the exact Zoeppritz's equation. The linear operator is a linear perturbation equation based on the elastic inverse-scattering theory. Due to the difference of angle gathers in adjacent traces, the linear forward operator may cause a deviation of the updated parameters. By comparison, the exact Zoeppritz's equation as the nonlinear forward operator has better applicability and precision. Based on the proposed method, the elastic parameters are obtained from seismic data. Numerical examples show that the inverted elastic parameters of the proposed method have a higher horizontal resolution, and the details in the inversion profile can be better highlighted.

This article is a continuation of our first work \cite{chaudruraynal:frikha}. We here establish some new quantitative estimates for propagation of chaos of non-linear stochastic differential equations in the sense of McKean-Vlasov. We obtain explicit error estimates, at the level of the trajectories, at the level of the semi-group and at the level of the densities, for the mean-field approximation by systems of interacting particles under mild regularity assumptions on the coefficients. A first order expansion for the difference between the densities of one particle and its mean-field limit is also established. Our analysis relies on the well-posedness of classical solutions to the backward Kolmogorov partial differential equations defined on the strip $[0,T] \times \mathbb{R}^d \times \mathcal{P}_2(\mathbb{R}^d)$, $\mathcal{P}_2(\mathbb{R}^d)$ being the Wasserstein space, that is, the space of probability measures on $\mathbb{R}^d$ with a finite second-order moment and also on the existence and uniqueness of a fundamental solution for the related parabolic linear operator here stated on $[0,T]\times \mathcal{P}_2(\mathbb{R}^d)$.; Cet article est la suite de notre premier travail \cite{chaudruraynal:frikha}. Nous \'etablissons ici de nouvelles estimations quantitatives pour la propagation du chaos des \'equations diff\'erentielles stochastiques non-lin\'eaires au sens de McKean-Vlasov. Nous obtenons des estimations d'erreurs explicites, au niveau des trajectoires, au niveau du semi-groupe et au niveau des densit\'es de transition, pour l'approximation champ moyen par des syst\`emes de particules en interaction sous de faibles hypoth\`eses de r\'egularit\'e sur les coefficients. Un d\'eveloppement \`a l'ordre un pour la diff\'erence entre les densit\'es d'une particule et celle de sa limite champ moyen est \'egalement \'etabli. Notre analyse repose sur le caract\`ere bien pos\'e de solutions classiques aux \'equations aux d\'eriv\'ees partielles de Kolmogorov r\'etrogrades d\'efinies sur la bande $[0,T] \times \mathbb{R}^d \times \mathcal{P}_2(\mathbb{R}^d)$, $\mathcal{P}_2(\mathbb{R}^d)$ \'etant l'espace de Wasserstein, c'est-\`a-dire l'espace des mesures de probabilit\'es sur $\mathbb{R}^d$ de moment d'ordre deux fini et aussi sur l'existence et l'unicit\'e d'une solution fondamentale pour l'op\'erateur parabolique lin\'eaire associ\'e \'enonc\'e ici sur $[0,T]\times \mathcal{P}_2(\mathbb{R}^d)$.

In many scientific fields, partly interval-censored data, which consist of exactly observed and interval-censored observations on the failure time of interest, appear frequently. However, methodological developments in the analysis of partly interval-censored data are relatively limited and have mainly focused on additive or proportional hazards models. The general linear transformation model provides a highly flexible modeling framework that includes several familiar survival models as special cases. Despite such nice features, the inference procedure for this class of models has not been developed for partly interval-censored data. We propose a fully Bayesian approach coped with efficient Markov chain Monte Carlo methods to fill this gap. A four-stage data augmentation procedure is introduced to tackle the challenges presented by the complex model and data structure. The proposed method is easy to implement and computationally attractive. The empirical performance of the proposed method is evaluated through two simulation studies, and the model is then applied to a dental health study.

This paper studies the leader-following state consensus problem for heterogeneous linear multi-agent systems under fixed directed communication topologies. First, we propose a consensus protocol consisting of four parts for high-order multi-agent systems, in which different agents are allowed to have different gain matrices so as to increase the degree of design freedom. Then, we adopt a state linear transformation, which is constructed based on the incidence matrix of a directed spanning tree of the communication topology, to equivalently transform the state consensus problem into a partial variable stability problem. Meanwhile, the results of the partial variable stability theory are used to derive a sufficient and necessary consensus criterion, expressed as the Hurwitz stability of a real matrix. Then, this criterion is further expressed as a bilinear matrix inequality condition, and, based on this condition, an iterative algorithm is proposed to find the gain matrices of the protocol. Finally, numerical examples are provided to verify the effectiveness of the proposed protocol design method.

Given a linear operator T on a Hilbert space, the pseudo-point spectrum, the pseudo-residual spectrum, and the pseudo-continuous spectrum of T are defined, and some of their properties are studied....

A method for recursive subspace identification of continuous-time systems based on generalized Poisson moment functionals is proposed. Most of the existing subspace identification methods have concentrated mainly on the time-invariant discrete-time systems. The results of subspace identification methods are confined to the discrete-time cases, due to the difference on the construction of Hankel matrices. In addition, the time-invariant identification algorithms are not suitable for online identification cases. In order to solve the problems above, the time derivatives of Hankel matrices can be evaluated by generalized Poisson moment functionals, which provides a simple linear mapping for identification algorithm without the amplification of stochastic noises. The size of the data matrices is fixed a priori to fade the influence of old data to the updated data, which is a key to reduce computational burden and storage cost of recursive algorithms. The efficiency of the presented method is provided by comparing simulation results.

Using a nonlocal version of the center manifold theorem and a normal form reduction, we prove the existence of small-amplitude generalized solitary-wave solutions and modulated solitary-wave solutions to the steady gravity-capillary Whitham equation with weak surface tension. Through the application of the center manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four-dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravity-capillary water wave problem, the associated linear operator is seen to undergo either a reversible $0^{2+}(i k_0)$ bifurcation or a reversible $(i s)^2$ bifurcation. Through a normal form transformation, the reduced system of ODEs along each relevant parameter curve is seen to be well approximated by a truncated system retaining only second-order or third-order terms. These truncated systems relate directly to systems obtained in the study of the full gravity-capillary water wave equation and, as such, the existence of generalized and modulated solitary waves for the truncated systems is guaranteed by classical works, and they are readily seen to persist as solutions of the gravity-capillary Whitham equation due to reversibility. Consequently, this work illuminates further connections between the gravity-capillary Whitham equation and the full two-dimensional gravity-capillary water wave problem., 35 pages, 4 figures. Some mathematical typos fixed

In Pasalic et al. (2016) a construction allowing for high levels of modification was presented. It can be used to construct several important combinatorial structures, among them are examples of complete permutations. Here the method is used to construct infinite classes of generalised complete permutations F ( x ) , where both F ( x ) and F ( x ) + h D ( x ) are permutations, not just F ( x ) + x . In the article three families of the function h D ( x ) are considered: h D ( x ) being a function multiplying the vector x with a vector D , a permutation matrix D , or a linear mapping matrix D . Most existing results related to complete permutations use the finite field notation, while in this article we are developing permutations based on the vector space structure. The case where D is a binary vector needs to be emphasised. In this case the permutation F ( x ) is defined in such a way that F ( x ) + D T x remains a permutation for any of the 2 n − 3 vectors D as defined in Eq. (5). Let S be the set of all such vectors. We present for arbitrary n > 4 construction of an S -complete permutation F ( x ) , where | S | = 2 n − 3 . Additionally, we prove that each of these permutations F has such a corresponding linear subspace L that the pair ( F , L ) satisfies the ( C ) -property and can be used to construct a huge infinite class of bent functions in Carlet’s C class. In Mandal et al. (2016) it was proven that finding such pairs ( F , L ) is a difficult problem.

This paper proposes a new procedure to build factor models for high-dimensional unit-root time series by postulating that a $p$-dimensional unit-root process is a nonsingular linear transformation of a set of unit-root processes, a set of stationary common factors, which are dynamically dependent, and some idiosyncratic white noise components. For the stationary components, we assume that the factor process captures the temporal-dependence and the idiosyncratic white noise series explains, jointly with the factors, the cross-sectional dependence. The estimation of nonsingular linear loading spaces is carried out in two steps. First, we use an eigenanalysis of a nonnegative definite matrix of the data to separate the unit-root processes from the stationary ones and a modified method to specify the number of unit roots. We then employ another eigenanalysis and a projected principal component analysis to identify the stationary common factors and the white noise series. We propose a new procedure to specify the number of white noise series and, hence, the number of stationary common factors, establish asymptotic properties of the proposed method for both fixed and diverging $p$ as the sample size $n$ increases, and use simulation and a real example to demonstrate the performance of the proposed method in finite samples. We also compare our method with some commonly used ones in the literature regarding the forecast ability of the extracted factors and find that the proposed method performs well in out-of-sample forecasting of a 508-dimensional PM$_{2.5}$ series in Taiwan., Comment: 45 pages, 11 figures. arXiv admin note: text overlap with arXiv:1808.07932

F 2 -linear pseudorandom number generators are very popular due to their high speed, to the ease with which generators with a sizable state space can be created, and to their provable theoretical properties. However, they suffer from linear artifacts that show as failures in linearity-related statistical tests such as the binary-rank and the linear-complexity test. In this article, we give two new contributions. First, we introduce two new F 2 -linear transformations that have been handcrafted to have good statistical properties and at the same time to be programmable very efficiently on superscalar processors, or even directly in hardware. Then, we describe some scramblers , that is, nonlinear functions applied to the state array that reduce or delete the linear artifacts, and propose combinations of linear transformations and scramblers that give extremely fast pseudorandom number generators of high quality. A novelty in our approach is that we use ideas from the theory of filtered linear-feedback shift registers to prove some properties of our scramblers, rather than relying purely on heuristics. In the end, we provide simple, extremely fast generators that use a few hundred bits of memory, have provable properties, and pass strong statistical tests.

This work studies the kernel of a linear operator associated with the generalized k-fold commutator. Given a set $\mathfrak{A}= \left\{ A_{1}, \ldots ,A_{k} \right\}$ of real $n \times n$ matrices, the commutator is denoted by$[A_{1}| \ldots |A_{k}]$. For a fixed set of matrices $\mathfrak{A}$ we introduce a multilinear skew-symmetric linear operator $T_{\mathfrak{A}}(X)=T(A_{1}, \ldots ,A_{k})[X]=[A_{1}| \ldots |A_{k} |X] $. For fixed $n$ and $k \ge 2n-1, \; T_{\mathfrak{A}} \equiv 0$ by the Amitsur--Levitski Theorem [2] , which motivated this work. The matrix representation $M$ of the linear transformation $T$ is called the k-commutator matrix. $M$ has interesting properties, e.g., it is a commutator; for $k$ odd, there is a permutation of the rows of $M$ that makes it skew-symmetric. For both $k$ and $n$ odd, a provocative matrix $\mathcal{S}$ appears in the kernel of $T$. By using the Moore--Penrose inverse and introducing a conjecture about the rank of $M$, the entries of $\mathcal{S}$ are shown to be quotients of polynomials in the entries of the matrices in $\mathfrak{A}$. One case of the conjecture has been recently proven by Brassil. The Moore--Penrose inverse provides a full rank decomposition of $M$.

From an APOS (Action–Process–Object–Schema) theory perspective, learning mathematics involves construction of knowledge through mental mechanisms, which evolves between different mental structures ...

The lack of interpretability of the neural network algorithm has become the bottleneck of its wide application. We propose a general mathematical framework, which couples the complex structure of the system with the nonlinear activation function to explore the decoupled dimension reduction method of high-dimensional system and reveal the calculation mechanism of the neural network. We apply our framework to some network models and a real system of the whole neuron map of Caenorhabditis elegans. Result shows that a simple linear mapping relationship exists between network structure and network behavior in the neural network with high-dimensional and nonlinear characteristics. Our simulation and theoretical results fully demonstrate this interesting phenomenon. Our new interpretation mechanism provides not only the potential mathematical calculation principle of neural network but also an effective way to accurately match and predict human brain or animal activities, which can further expand and enrich the interpretable mechanism of artificial neural network in the future.

Made available in DSpace on 2022-05-01T08:45:02Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-01-01 We investigate several notions related to pseudotrajectories, including chain recurrence and shadowing properties, for a special class of diffeomorphisms on euclidean spheres, known as spherical linear transformations, and for bounded linear operators on Banach spaces. Our main results are complete characterizations of chain recurrence for spherical linear transformations on euclidean spheres and for weighted shifts on the classical Banach sequence spaces c and ℓp. Another main result is a characterization of hyperbolicity for invertible operators on Banach spaces by means of average expansivity and the average shadowing property. Instituto Federal de Educação Ciência e Tecnologia de São Paulo Campus Presidente Epitácio, Rua José Ramos Junior, 27-50 Departamento de Matemática Aplicada Instituto de Matemática Universidade Federal do Rio de Janeiro, Caixa Postal 68530 Departamento de Matemática Universidade Estadual Paulista, Rua Cristóvão Colombo, 2265, Jardim Nazareth Departamento de Matemática Universidade Estadual Paulista, Rua Cristóvão Colombo, 2265, Jardim Nazareth

In this paper, we give a characterization of a B-Fredholm quotient of two bounded linear operators A and M satisfying certain conditions on a Hilbert space. We derive also the relationships between some B-Fredholm spectra of the linear operator pencil $$\lambda M-A$$ , where $$\lambda \in {\mathbb {C}}$$ , and those of the quotient of A and M. These results are finally applied to a closed densely defined linear operator $$T=\Gamma (C)=C/(I-C^{*}C)^{\frac{1}{2}}$$ , where C a pure quasi-normal contraction and the symbol I the identity operator, by giving a B-Fredholm correspondence between T and C, and some related results concerning B-Weyl operators and Drazin invertible ones.

In this paper we intend to describe generalized Lie-type derivations using, among other things, a generalization for alternative algebras of the result: “If $F:A\to A$ is a generalized Lie n-derivation associated with a Lie n-derivation D, then a linear map $H=F-D$ satisfies $H(p_n(x_1,x_2,\ldots ,x_n)) =p_n(H(x_1),x_2,\ldots ,x_n)$ for all $x_1,x_2,\ldots ,x_n\in A$ ”. Thus, if A is a unital alternative algebra with a nontrivial idempotent e1 satisfying certain conditions, then a generalized Lie-type derivation $F : A \rightarrow A$ is of the form $F(x) = \lambda x + \Xi(x)$ for all $x \in A$ , where $\lambda \in Z(A)$ and $\Xi : A \rightarrow A$ is a Lie-type derivation.