We develop Jacobson's refinement of Engel's Theorem for Leibniz algebras. We then note some consequences of the result.Since Leibniz algebras were introduced in [Loday 1993] as a noncommutative generalization of Lie algebras, one theme has been to extend Lie algebra results to Leibniz algebras. In particular, Engel's theorem has been extended in [Ayupov and Omirov 1998;Barnes 2011;Patsourakos 2007]. In the second of these works, the classical Engel's theorem is used to give a short proof of the result for Leibniz algebras. The proofs in the other two papers do not use the classical theorem and, therefore, the Lie algebra result is included in the result. In this note, we give two proofs of the generalization to Leibniz algebras of Jacobson's refinement to Engel's theorem, a short proof which uses Jacobson's theorem and a second proof which does not use it. It is interesting to note that the technique of reducing the problem to the special Lie algebra case significantly shortens the proof for the general Leibniz algebras case. This approach has been used in a number of situations [Barnes 2011]. We also note some standard consequences of this theorem. The proofs of the corollaries are exactly as in Lie algebras (see [Kaplansky 1971]). Our result can be used to directly show that the sum of nilpotent ideals is nilpotent, and hence one has a nilpotent radical. In this paper, we consider only finite dimensional algebras and modules over a field ‫.ކ‬ An algebra A is called Leibniz if it satisfies x(yz) = (x y)z + y(x z). Denote by R a and L a , respectively, right and left multiplication by a ∈ A. Then Using (1) and (2) we obtainwhere a 1 = a and a n is defined inductively as a n+1 = aa n . Furthermore, for n > 1,For any set X in an algebra, we let X denote the algebra generated by X .is a combination of terms with each term having at least 2n − 1 factors. Moreover, each of these factors is either L a or R a . Any L a to the right of the first R a can be turned into an R a using (4). Hence, any term with 2n − 1 factors can be converted into a term with either L a in the first n − 1 leading positions or R a in the last n postitions. In either case, the term is 0 and s 2n−1 = 0. Thus R a , L a is nil and hence nilpotent.Let M be an A-bimodule and let T a (m) = am and S a (m) = ma, a ∈ A, m ∈ M. The analogues of (1)- (4) hold:These operations have the same properties as L a and R a , and the associative algebra T a , S a generated by all T b , S b , b ∈ a is nilpotent if T a is nilpotent. We record this as Lemma. Let A be a finite dimensional Leibniz algebra and let a ∈ A. Let M be a finite dimensional A-bimodule such that T a is nilpotent on M. Then S a is nilpotent, and S a , T a , the algebra generated by all S b , T b , b ∈ a , is nilpotent.A subset of A which is closed under multiplication is called a Lie set.Theorem (Jacobson's refinement of Engel's theorem for Leibniz algebras). Let A be a finite dimensional Leibniz algebra and M be a finite dimensional A-bimodule. Let C be a Lie set in A such that A = C . Suppose...

We develop Jacobson's refinement of Engel's Theorem for Leibniz algebras. We then note some consequences of the result.Since Leibniz algebras were introduced by Loday in [6] as a noncommutative generalization of Lie algebras, one theme is to extend Lie algebra results to Leibniz algebras. In particular, Engel's theorem has been extended in [1], [3], and [7]. In [3], the classical Engel's theorem is used to give a short proof of the result for Leibniz algebras. The proofs in [1] and [7] do not use the classical theorem and, therefore, the Lie algebra result is included in the result. In this note, we give two proofs of the generalization to Leibniz algebras of Jacobson's refinement to Engel's theorem, a short proof which uses Jacobson's theorem and a second proof which does not use it. It is interesting to note that the technique of reducing the problem to the special Lie algebra case significantly shortens the proof for the general Leibniz algebras case. This approach has been used in a number of situations, see [2]. We also note some standard consequences of this theorem. The proofs of the corollaries are exactly as in Lie algebras (see [5]). Our result can be used to directly show that the sum of nilpotent ideals is nilpotent, and hence one has a nilpotent radical. In this paper, we consider only finite dimensional algebras and modules over a field F.An algebra A is called Leibniz if it satisfies x(yz) = (xy)z + y(xz). Denote by R a and L a , respectively, right and left multiplication by

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Symmetric k-varieties generalize Riemannian symmetric spaces to reductive groups defined over arbitrary fields. For most perfect fields, it is known that symmetric k-varieties are in one-to-one correspondence with isomorphy classes of k-involutions. Therefore, it is useful to have representatives of each isomorphy class in order to describe the k-varieties. Here we give matrix representatives for each isomorphy class of k-involutions of SL(n, k) in the case that k is any field of characteristic 2; we also describe fixed point groups of each type of involution.

Introduction: There are several recommendations advising caregivers when and how to introduce solid food to infants. These complementary feeding guides vary in terms of the recommendations for timing and portions. The objective of this study is to determine the impact of following different guidelines on weight trajectories of infants. Methods:In 2018, the study team developed a computational simulation model to capture feeding behaviors, activity levels, metabolism, and body size of infants from 6 months to 1 year. Daily food intake of virtual infants based on feeding recommendations translated to changes in body weight. Next, simulations tested the impact of the following complementary feeding recommendations that provided amount, type, and timing of foods: Children's Hospital of Philadelphia, Johns Hopkins Medicine, Enfamil, and Similac.Results: When virtual caregivers fed infants according to the four different guides, none of the simulated situations resulted in normal weight at 12 months when infants were also being breastfed along average observed patterns. Reducing breast milk portions in half while caregivers fed infants according to complementary feeding guidelines resulted in overweight BMIs between 9 and 11 months for Children's Hospital of Philadelphia, Johns Hopkins Medicine, and Enfamil guidelines. Cutting breast milk portions in half also led to infants reaching unhealthy underweight BMI percentiles between 7 and 11 months for female and male infants when caregivers followed Children's Hospital of Philadelphia, Johns Hopkins Medicine, and Similac guidelines.Conclusions: This study identified situations in which infants could reach unhealthy weights, even while following complementary feeding guidelines, suggesting that current recommended portion sizes should be tightened.