Derivations Of Algebras Research Articles

Interacting Quantum Atoms-A Review.

The aim of this review is threefold. On the one hand, we intend it to serve as a gentle introduction to the Interacting Quantum Atoms (IQA) methodology for those unfamiliar with it. Second, we expect it to act as an up-to-date reference of recent developments related to IQA. Finally, we want it to highlight a non-exhaustive, yet representative set of showcase examples about how to use IQA to shed light in different chemical problems. To accomplish this, we start by providing a brief context to justify the development of IQA as a real space alternative to other existent energy partition schemes of the non-relativistic energy of molecules. We then introduce a self-contained algebraic derivation of the methodological IQA ecosystem as well as an overview of how these formulations vary with the level of theory employed to obtain the molecular wavefunction upon which the IQA procedure relies. Finally, we review the several applications of IQA as examined by different research groups worldwide to investigate a wide variety of chemical problems.

Homogeneous Einstein manifolds based on symplectic triple systems

Abstract For each simple symplectic triple system over the real numbers, the standard enveloping Lie algebra and the algebra of inner derivations of the triple provide a reductive pair related to a semi-Riemannian homogeneous manifold. It is proved that this is an Einstein manifold.

Almost inner derivations of 2-step nilpotent Lie algebras of genus 2

We study almost inner derivations of 2-step nilpotent Lie algebras of genus 2, i.e., having a 2-dimensional commutator ideal, using matrix pencils. In particular we determine all almost inner derivations of such algebras in terms of minimal indices and elementary divisors over an arbitrary algebraically closed field of characteristic not 2 and over the real numbers.

A novel image encryption method based on fractional discrete Meixner moments

In order to improve the performance of the Meixner moments, we suggest in this article a new set of Fractional Discrete Meixner Polynomials (FrDMPs), a sort of a generalization of the traditional whole order. Firstly, we introduce the necessary algebraic derivatives of FrDMPs using the spectral decomposition of Discrete Meixner Polynomials (DMPs) for any order without numerical fluctuation and always preserving the property of orthogonality thanks to the algorithm of the Gram-Schmidt process. Then, we determine the eigenvalues and the corresponding multiplicity of the Meixner transform matrix. An image encryption and decryption method was proposed based on jigsaw transform and generation of Fractional Discrete Meixner Moments (FrDMMs), in which the image is broken up into bit planes. Each bit plane undergoes a jigsaw transform is divided into blocks and encrypted. The transformed bit planes are combined together and then encrypted using random phase masks and fractional discrete Meixner moments. In addition, we introduce a new encryption and decryption scheme based on the proposed FrDMMs. This scheme has a good encryption effect because the fractional parameters used as key to the encrypted data. Experimental results show that the encryption scheme is sensitive to keys, it is resist a variety of attacks, and decrypted images are almost non-distored, which indicate excellent encryption effect, sufficient security and robustness of the method.

On the non-existence of negative weight derivations of the new moduli algebras of singularities

We propose a new conjecture about the non-existence of negative weight derivations of the new moduli algebras of weighted homogeneous hypersurface singularities and verify this conjecture up to dimension three.

Aristotle’s geometrical accounting

Abstract This paper comments on Ambrosi’s ‘Aristotle’s geometrical accounting’ (Cambridge Journal of Economics 2018, 42, 543–576). While supporting Ambrosi’s case that Aristotle may well have used geometry to demonstrate accounting equivalence, the paper takes issue with Ambrosi’s specific approach to interpreting Aristotle’s ‘merit or worth’ in terms of income. It is entirely reasonable to understand worth in terms of income. However, Ambrosi implies that worth is determined by market prices; this is inconsistent with Aristotle’s treatment of distributive justice. Rather Aristotle states that relative worth is determined prior to market exchange. In other words, Aristotle seeks to identify the just price to which market prices may or may not correspond. Ambrosi’s own geometrical method is extended to illustrate this point, to provide a geometrical alternative to Ambrosi’s algebraic derivation of the key proposition ‘as builder to shoemaker, so shoes to houses’, and to resolve what Aristotle means by ‘one should not introduce them as terms in a figure of proportion when they are already making the exchange (since otherwise one of the two extremes will have both of the excess amounts) but when they are still in possession of their own products’. Aristotle deserves recognition as the first Western economist and as a Classical economist to boot, in the sense of holding that distribution is determined prior to exchange.

On the Generalized Cartan Matrices Arising from k-th Yau Algebras of Isolated Hypersurface Singularities

Let (V,0) be an isolated hypersurface singularity defined by the holomorphic function $f: (\mathbb {C}^{n}, 0)\rightarrow (\mathbb {C}, 0)$ . The k-th Yau algebra Lk(V ) is defined to be the Lie algebra of derivations of the k-th moduli algebra $A^{k}(V) := \mathcal {O}_{n}/(f, m^{k}J(f))$ , where k ≥ 0, m is the maximal ideal of $\mathcal {O}_{n}$ . I.e., Lk(V ) := Der(Ak(V ),Ak(V )). These new series of derivation Lie algebras are quite subtle invariants since they capture enough information about the complexity of singularities. In this paper we formulate a conjecture for the complete characterization of ADE singularities by using generalized Cartan matrix Ck(V ) associated to k-th Yau algebras Lk(V ), k ≥ 1. In this paper, we provide evidence for the conjecture and give a new complete characterization for ADE singularities. Furthermore, we compute their other various invariants that arising from the 1-st Yau algebra L1(V ).

Lie algebras of vertical derivations on semiaffine varieties with torus actions

Let X be a normal variety endowed with an algebraic torus action. An additive group action α on X is called vertical if a general orbit of α is contained in the closure of an orbit of the torus action and the image of the torus normalizes the image of α in Aut(X). Our first result in this paper is a classification of vertical additive group actions on X under the assumption that X is proper over an affine variety. Then we establish a criterion as to when the infinitesimal generators of a finite collection of additive group actions on X generate a finite-dimensional Lie algebra inside the Lie algebra of derivations of X.

On two inequality conjectures for the k-th Yau numbers of isolated hypersurface singularities

Let (V, 0) be an isolated hypersurface singularity defined by the holomorphic function $$f: ({\mathbb {C}}^n, 0)\rightarrow ({\mathbb {C}}, 0)$$ . In our previous work, we introduced a series of novel Lie algebras associated to (V, 0), i.e., k-th Yau algebra $$L^k(V), k\ge 0$$ . It was defined to be the Lie algebra of derivations of the k-th moduli algebras $$A^k(V)= {\mathcal {O}}_n/(f, m^k J(f)), k\ge 0$$ , where m is the maximal ideal of $${\mathcal {O}}_n$$ . I.e., $$L^k(V):=\text {Der}(A^k(V), A^k(V))$$ . The dimension of $$L^k(V)$$ was denoted by $$\lambda ^k(V)$$ . The number $$\lambda ^k(V)$$ , which was called k-th Yau number, is a subtle numerical analytic invariant of (V, 0). Furthermore, we formulated two conjectures for these k-th Yau number invariants: a sharp upper estimate conjecture of $$\lambda ^k(V)$$ for weighted homogeneous isolated hypersurface singularities (see Conjecture 1.2) and an inequality conjecture $$\lambda ^{(k+1)}(V) > \lambda ^k(V), k\ge 0$$ (see Conjecture 1.1). In this article, we verify these two conjectures when k is small for large class of singularities.

Finite-temperature many-body perturbation theory in the grand canonical ensemble.

A finite-temperature many-body perturbation theory is presented, which expands in power series the electronic grand potential, chemical potential, internal energy, and entropy on an equal footing. Sum-over-states and sum-over-orbitals analytical formulas for the second-order perturbation corrections to these thermodynamic properties are obtained in a time-independent, nondiagrammatic, algebraic derivation, relying on the sum rules of the Hirschfelder-Certain degenerate perturbation energies in a degenerate subspace as well as nine algebraic identities for the zeroth-order thermal averages of one- through four-indexed quantities and products thereof. They reproduce numerically exactly the benchmark data obtained as the numerical derivatives of the thermal-full-configuration-interaction results for a wide range of temperatures.

Lie-central derivations, Lie-centroids and Lie-stem Leibniz algebras

In this paper, we introduce the notion Lie-derivation. This concept generalizes derivations for non-Lie Leibniz algebras. We study these Lie-derivations in the case where their image is contained in the Lie-center, call them Lie-central derivations. We provide a characterization of Lie-stem Leibniz algebras by their Lie-central derivations, and prove several properties of the Lie algebra of Lie-central derivations for Lie-nilpotent Leibniz algebras of class 2. We also introduce ${\sf ID}_*-Lie$-derivations. A ${\sf ID}_*-Lie$-derivation of a Leibniz algebra G is a Lie-derivation of G in which the image is contained in the second term of the lower Lie-central series of G, and that vanishes on Lie-central elements. We provide an upperbound for the dimension of the Lie algebra $ID_*^{Lie}(G)$ of $ID_*Lie$-derivation of G, and prove that the sets $ID_*^{Lie}(G)$ and $ID_*^{Lie}(G)$ are isomorphic for any two Lie-isoclinic Leibniz algebras G and Q.

Lie Algebras of Heat Operators in a Nonholonomic Frame

We construct the Lie algebras of systems of $$2g$$ graded heat operators $$Q_0,Q_2,\dots,Q_{4g-2}$$ that determine the sigma functions $$\sigma(z,\lambda)$$ of hyperelliptic curves of genera $$g=1$$ , $$2$$ , and $$3$$ . As a corollary, we find that the system of three operators $$Q_0$$ , $$Q_2$$ , and $$Q_4$$ is already sufficient for determining the sigma functions. The operator $$Q_0$$ is the Euler operator, and each of the operators $$Q_{2k}$$ , $$k>0$$ , determines a $$g$$ -dimensional Schrodinger equation with potential quadratic in $$z$$ for a nonholonomic frame of vector fields in the space $$\mathbb C^{2g}$$ with coordinates $$\lambda$$ . For any solution $$\varphi(z,\lambda)$$ of the system of heat equations, we introduce the graded ring $$\mathscr R_\varphi$$ generated by the logarithmic derivatives of $$\varphi(z,\lambda)$$ of order $$\ge 2$$ and present the Lie algebra of derivations of $$\mathscr R_\varphi$$ explicitly. We show how this Lie algebra is related to our system of nonlinear equations. For $$\varphi(z,\lambda)=\sigma(z,\lambda)$$ , this leads to a well-known result on how to construct the Lie algebra of differentiations of hyperelliptic functions of genus $$g=1,2,3$$ .

On the characterization of the space of derivations in evolution algebras

We study the space of derivations for some finite-dimensional evolution algebras, depending on the twin partition of an associated directed graph. For evolution algebras with a twin-free associated graph, we prove that the space of derivations is zero. For the remaining families of evolution algebras, we obtain sufficient conditions under which the study of such a space can be simplified. We accomplish this task by identifying the null entries of the respective derivation matrix. Our results suggest how strongly the associated graph’s structure impacts in the characterization of derivations for a given evolution algebra. Therefore, our approach constitutes an alternative to the recent developments in the research of this subject. As an illustration of the applicability of our results, we provide some examples and we exhibit the classification of the derivations for non-degenerate irreducible three-dimensional evolution algebras.

Derivations of Group Algebras

In the paper, a method of describing the outer derivations of the group algebra of a finitely presentable group is given. The description of derivations is given in terms of characters of the groupoid of the adjoint action of the group.

Nilpotent Lie algebras of derivations with the center of small corank

Let $\mathbb K$ be a field of characteristic zero, $A$ be an integral domain over $\mathbb K$ with the field of fractions $R=Frac(A),$ and $Der_{\mathbb K}A$ be the Lie algebra of all $\mathbb K$-derivations on $A$. Let $W(A):=RDer_{\mathbb K} A$ and $L$ be a nilpotent subalgebra of rank $n$ over $R$ of the Lie algebra $W(A).$ We prove that if the center $Z=Z(L)$ is of rank $\geq n-2$ over $R$ and $F=F(L)$ is the field of constants for $L$ in $R,$ then the Lie algebra $FL$ is contained in a locally nilpotent subalgebra of $ W(A)$ of rank $n$ over $R$ with a natural basis over the field $R.$ It is also proved that the Lie algebra $FL$ can be isomorphically embedded (as an abstract Lie algebra) into the triangular Lie algebra $u_n(F)$, which was studied early by other authors.

Local and 2-local derivations of solvable Leibniz algebras

We show that any local derivation on the solvable Leibniz algebras with model or abelian nilradicals, whose dimension of complementary space is maximal is a derivation. We show that solvable Leibniz algebras with abelian nilradicals, which have [Formula: see text] dimension complementary space, admit local derivations which are not derivations. Moreover, similar problem concerning [Formula: see text]-local derivations of such algebras is investigated and an example of solvable Leibniz algebra is given such that any [Formula: see text]-local derivation on it is a derivation, but which admits local derivations which are not derivations.

Imbalance-XGBoost: leveraging weighted and focal losses for binary label-imbalanced classification with XGBoost

The paper presents Imbalance-XGBoost, a Python package that combines the powerful XGBoost software with weighted and focal losses to tackle binary label-imbalanced classification tasks. Though a small-scale program in terms of size, the package is, to the best of our knowledge, the first of its kind which provides an integrated implementation for the two loss functions on XGBoost and brings a general-purpose extension to XGBoost for label-imbalanced scenarios. In this paper, the design and usage of the package are discussed and illustrated with examples. Furthermore, as the first- and second-order derivatives of the loss functions are essential for the implementations, the algebraic derivation is discussed and it can be deemed as a separate contribution. The performances of the methods implemented in the package are extensively evaluated on Parkinson’s disease classification dataset, and multiple competitive performances are presented with the ROC and Precision-Recall (PR) curves. To further assert the superiority of the methods, the performances on four other benchmark datasets from the UCI machine learning repository are additionally reported. Given the scalable nature of XGBoost, the package has great potentials to be broadly applied to real-life binary classification tasks, which are usually of large-scale and label-imbalanced.

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras

Commutative post-Lie algebra structures on Lie algebras, in short CPA structures, have been studied over fields of characteristic zero, in particular for real and complex numbers motivated by geometry. A perfect Lie algebra in characteristic zero only admits the trivial CPA-structure. In this article we study these structures over fields of characteristic p>0. We show that every perfect modular Lie algebra in characteristic p>2 having a solvable outer derivation algebra admits only the trivial CPA-structure. This involves a conjecture by Hans Zassenhaus, saying that the outer derivation algebra Out(g) of a simple modular Lie algebra g is solvable. We try to summarize the known results on the Zassenhaus conjecture and prove some new results using the classification of simple modular Lie algebras by Premet and Strade for algebraically closed fields of characteristic p>3. As a corollary we obtain that every central simple modular Lie algebra of characteristic p>3 admits only the trivial CPA-structure.

Monotonic convergence and robustness of higher‐order gain‐adaptive iterative learning control

SummaryFor a class of repetitive linear discrete time‐invariant systems with higher relative degree, a higher‐order gain‐adaptive iterative learning control (HOGAILC) is developed while minimizing the energy increment of two adjacent tracking errors with the argument being the iteration‐time‐variable learning‐gain vector (ITVLGV). By taking advantage of rows/columns exchanging transformation of matrix, the ITVLGV is achieved in an explicit form which is dependent upon the system Markov parameters and adaptive to the iterationwise tracking‐error vector. Algebraic derivation demonstrates that the HOGAILC is strictly monotonously convergent. On the basis of the adaptive mode, a damping quasi‐HOGAILC strategy is exploited while the uncertainties of the system Markov parameters exist. Rigorous analysis delivers that the damping quasi‐scheme is strictly monotonically convergent and thus the HOGAILC mechanism is robust to a wider range of uncertainty of system parameters and the damping factor may relax the uncertainty range. Numerical simulations are made to illustrate the validity and the effectiveness.

On classification of 2-dimensional evolution algebras and its applications

In the paper we give a complete classification of 2-dimensional evolution algebras over algebraically closed fields, we compare the list of representatives of the isomorphism classes with that of obtained earlier by the other authors. Also we describe their groups of automorphisms and derivation algebras.