Search results for: leibniz n-algebras

Authors: Natalia Pacheco Rego

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The Liezation functor is a map from the category of Leibniz algebras to the category of Lie algebras, which assigns a Leibniz algebra to the Lie algebra given by the quotient of the Leibniz algebra by the ideal spanned by the square elements of the Leibniz algebra. This functor is left adjoint to the inclusion functor that considers a Lie algebra as a Leibniz algebra. This environment fits in the framework of central extensions and commutators in semi-abelian categories with respect to a Birkhoff subcategory, where classical or absolute notions are relative to the abelianization functor. Classical properties of Leibniz algebras (properties relative to the abelianization functor) were adapted to the relative setting (with respect to the Liezation functor); in general, absolute properties have the corresponding relative ones, but not all absolute properties immediately hold in the relative case, so new requirements are needed. Following this line of research, it was conducted an analysis of central derivations of Leibniz algebras relative to the Liezation functor, called as Lie-derivations, and a characterization of Lie-stem Leibniz algebras by their Lie-central derivations was obtained. In this paper, we present an overview of these results, and we analyze some new properties concerning Lie-central derivations and almost inner Lie-derivations. Namely, a Leibniz algebra is a vector space equipped with a bilinear bracket operation satisfying the Leibniz identity. We define the Lie-bracket by [x, y]lie = [x, y] + [y, x] , for all x, y . The Lie-center of a Leibniz algebra is the two-sided ideal of elements that annihilate all the elements in the Leibniz algebra through the Lie-bracket. A Lie-derivation is a linear map which acts as a derivative with respect to the Lie-bracket. Obviously, usual derivations are Lie-derivations, but the converse is not true in general. A Lie-derivation is called a Lie-central derivation if its image is contained in the Lie-center. A Lie-derivation is called an almost inner Lie-derivation if the image of an element x is contained in the Lie-commutator of x and the Leibniz algebra. The main results we present in this talk refer to the conditions under which Lie-central derivation and almost inner Lie-derivations coincide.

Keywords: almost inner Lie-derivation, Lie-center, Lie-central derivation, Lie-derivation

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Authors: Natália Maria Rego

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ALeibnizn-algebraGis aK-vector space endowed whit a n-linearbracket operation [-,…-] : GG … G→ Gsatisfying the fundamental identity, which can be expressed saying that the right multiplication map Ry2, …, ᵧₙ: Gn→ G, Rᵧ₂, …, ᵧₙn(ˣ¹, …, ₓₙ) = [[ˣ¹, …, ₓₙ], ᵧ₂, …, ᵧₙ], is a derivation. This structure, together with its skew-symmetric version, named as Lie n-algebra or Filippov algebra, arose in the setting of Nambumechanics, an n-ary generalization of the Hamiltonian mechanics. Thefirst goal of this work is to provide a characterization of various classes of central extensions of Leibniz n-algebras in terms of homological properties. Namely, Commutator extension, Quasi-commutator extension, Stem extension, and Stem cover. These kind of central extensions are characterized by means of the character of the map *(E): nHL1(G) → M provided by the five-term exact sequence in homology with trivial coefficients of Leibniz n-algebras associated to an extension E : 0 → M → K → G → 0. For a free presentation 0 →R→ F →G→ 0of a Leibniz n-algebra G,the term M(G) = (R[F,…n.., F])/[R, F,..n-1..,F] is called the Schur multiplier of G, which is a Baer invariant, i.e., it does not depend on the chosen free presentation, and it is isomorphic to the first Leibniz n-algebras homology with trivial coefficients of G. A central extension of Leibniz n-algebras is a short exact sequenceE : 0 →M→K→G→ 0such that [M, K,.. ⁿ⁻¹.., K]=0. It is said to be a stem extension if M⊆[G, .. n.., G]. Additionally, if the induced map M(K) → M(G) is the zero map, then the stem extension Eis said to be a stem cover. The second aim of this work is to analyze the interplay between stem covers of Leibniz n-algebras and the Schur multiplier. Concretely, in the case of finite-dimensional Leibniz n-algebras, we show the existence of coverings, and we prove that all stem covers with finite-dimensional Schur multiplier are isoclinic. Additionally, we characterize stem covers of perfect Leibniz n-algebras.

Keywords: leibniz n-algebras, central extensions, Schur multiplier, stem cover

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Authors: Remus Titiriga

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The paper focuses on Pascal's indivisibles evolving to Leibniz's infinitesimals. It starts with parallel developments by the two savants in Combinatorics (triangular numbers for Pascal and harmonic triangles for Leibniz) and their implication in determining the sum of mathematical series. It follows with a focus on the geometrical contributions of Pascal. He considered the cycloid and other mechanical curves the epitome of geometric comprehensibility in a series of challenging problems he posed to the mathematical world. Pascal provided the solutions in 1658, in a volume published under the pseudonym of Dettonville, using indivisibles and ratios between curved and straight lines. In the third part, the research follows the impact of this volume on Leibniz as the initial impetus for the elaboration of modern calculus as an algorithmic method disjoint of geometrical intuition. Then paper analyses the further steps and proves that Leibniz's developments relate to his philosophical frame (the search for a characteristic Universalis, the consideration of principle of continuity or the rule of sufficient reason) different from Pascal's and impacting mathematical problems and their solutions. At this stage in Leibniz's evolution, the infinitesimals replaced the indivisibles proper. The last part of the paper starts with speculation around "What if?". Could Pascal, if he lived more, accomplish the same feat? The document uses Pascal's reconstructed philosophical frame to formulate a positive answer. It also proposes to teach calculus with indivisibles and infinitesimals mimicking Pascal and Leibniz's achievements.

Keywords: indivisibles, infinitesimals, characteristic triangle, the principle of continuity

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Authors: Kreangkri Ratchagit

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This paper is concerned with exponential stability of switched linear systems with interval time-varying delays. The time delay is any continuous function belonging to a given interval, in which the lower bound of delay is not restricted to zero. By constructing a suitable augmented Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, a switching rule for the exponential stability of switched linear systems with interval time-varying delays and new delay-dependent sufficient conditions for the exponential stability of the systems are first established in terms of LMIs. Finally, some examples are exploited to illustrate the effectiveness of the proposed schemes.

Keywords: exponential stability, hybrid systems, timevarying delays, Lyapunov-Krasovskii functional, Leibniz-Newton’s formula

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Authors: Grienggrai Rajchakit

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This paper is concerned with the exponential stability of switched linear systems with interval time-varying delays. The time delay is any continuous function belonging to a given interval, in which the lower bound of delay is not restricted to zero. By constructing a suitable augmented Lyapunov-Krasovskii functional combined with Leibniz-Newton's formula, a switching rule for the exponential stability of switched linear systems with interval time-varying delays and new delay-dependent sufficient conditions for the exponential stability of the systems are first established in terms of LMIs. Finally, some examples are exploited to illustrate the effectiveness of the proposed schemes.

Keywords: exponential stability, hybrid systems, time-varying delays, lyapunov-krasovskii functional, leibniz-newton's formula

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Authors: Jan Busch, Peter Nyhuis

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Considering the challenges of short product life cycles and growing variant diversity, cost minimization and manufacturing flexibility increasingly gain importance to maintain a competitive edge in today’s global and dynamic markets. In this context, an aerodynamic part feeding system for high-speed industrial assembly applications has been developed at the Institute of Production Systems and Logistics (IFA), Leibniz Universitaet Hannover. The aerodynamic part feeding system outperforms conventional systems with respect to its process safety, reliability, and operating speed. In this paper, a multi-objective optimisation of the aerodynamic feeding system regarding the orientation rate, the feeding velocity and the required nozzle pressure is presented.

Keywords: aerodynamic feeding system, genetic algorithm, multi-objective optimization, workpiece orientation

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Authors: Jan Busch, Maurice Schmidt, Peter Nyhuis

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The change of conditions for production companies in high-wage countries is characterized by the globalization of competition and the transition of a supplier´s to a buyer´s market. The companies need to face the challenges of reacting flexibly to these changes. Due to the significant and increasing degree of automation, assembly has become the most expensive production process. Regarding the reduction of production cost, assembly consequently offers a considerable rationalizing potential. Therefore, an aerodynamic feeding system has been developed at the Institute of Production Systems and Logistics (IFA), Leibniz Universitaet Hannover. In former research activities, this system has been enabled to adjust itself using genetic algorithm. The longer the genetic algorithm is executed the better is the feeding quality. In this paper, the relation between the system´s setting time and the feeding quality is observed and a function which enables the user to achieve the minimum of the total feeding time is presented.

Keywords: aerodynamic feeding system, batch size, optimisation, setting time

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Authors: Sudhir Kumar Tewatia, Kanishck Tewatia, Anttriksh Tewatia

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Advancements in soil consolidation are discussed, and further improvements are proposed with particular reference to Tewatia’s Y-Y’ Approach, which is called the Settlement versus Rate of Settlement Approach in consolidation. A branch of calculus named Y-Y' (or y versus dy/dx) is suggested (as compared to the common X-Y', x versus dy/dx, dy/dx versus x or Newton-Leibniz branch) that solves some complicated/unsolved theoretical and practical problems in physical sciences (Physics, Chemistry, Mathematics, Biology, and allied sciences) and engineering in an amazingly simple and short manner, particularly when independent variable X is unknown and X-Y' Approach can’t be used. Complicated theoretical and practical problems in 1D, 2D, 3D Primary and Secondary consolidations with non-uniform gradual loading and irregularly shaped clays are solved with elementary school level Y-Y' Approach, and it is interesting to note that in X-Y' Approach, equations become more difficult while we move from one to three dimensions, but in Y-Y' Approach even 2D/3D equations are very simple to derive, solve, and use; rather easier sometimes. This branch of calculus will have a far-reaching impact on understanding and solving the problems in different fields of physical sciences and engineering that were hitherto unsolved or difficult to be solved by normal calculus/numerical/computer methods. Some particular cases from soil consolidation that basically creeps and diffusion equations in isolation and in combination with each other are taken for comparison with heat transfer. The Y-Y’ Approach can similarly be applied in wave equations and other fields wherever normal calculus works or fails. Soil mechanics uses mathematical analogies from other fields of physical sciences and engineering to solve theoretical and practical problems; for example, consolidation theory is a replica of the heat equation from thermodynamics with the addition of the effective stress principle. An attempt is made to give them mathematical analogies.

Keywords: calculus, clay, consolidation, creep, diffusion, heat, settlement

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