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The Product of Exponentials (PoE) formula is a mathematical tool that is used extensively in robotics. The virtue of using the exponential mapping, Lie Algebra and screw theory is that it allows an elegant and concise way of describing the…
A novel combinatorial formula is developed for for tensor product multiplicities in representation theory. We introduce a difference formula linking these multiplicities to restricted occupancy coefficients via a shifted operator. This…
In this paper we use some basic facts from the theory of (matrix) Lie groups and algebras to show that many of the classical matrix splittings used to construct stationary iterative methods and preconditioniers for Krylov subspace methods…
The operational formulations of quantum theory are drastically time oriented. However, to the best of our knowledge, microscopic physics is time-symmetric. We address this tension by showing that the asymmetry of the operational…
Prime-based ordering which is proved to be admissible, is the encoding of indeterminates in power-products with prime numbers and ordering them by using the natural number order. Using Eiffel, four versions of Buchberger's improved…
We investigate the solution to the logistic equation involving non-local operators in time. In the linear case such operators lead to the well-known theory of time changes. We provide the probabilistic representation for the non-linear…
In this paper, we define a new cohomology theory for multiplicative Hom-pre-Lie algebras which controls deformations of Hom-pre-Lie algebra structure. This new cohomology is a natural one by considering the structure map. We develop…
In this paper we introduce the concepts of a Rota-Baxter operator and a differential operator with weights on an $n$-algebra. We then focus on Rota-Baxter 3-Lie algebras and show that they can be derived from Rota-Baxter Lie algebras and…
Rota-Baxter associative algebras and Rota-Baxter Lie algebras are both important in mathematics and mathematical physics, with the former a basic structure in quantum field renormalization and the latter a operator form of the classical…
This work proposes a Bayesian inference method for the reduced-order modeling of time-dependent systems. Informed by the structure of the governing equations, the task of learning a reduced-order model from data is posed as a Bayesian…
In this paper, we introduce and study post-Lie conformal algebras (PLCAs), a generalization of post-Lie algebras to conformal algebras. We establish an equivalence between PLCA structures and Rota-Baxter operators of weight 1 on Lie…
In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce ON-structures on bimodules over pre-Lie algebras. We show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. We study…
This paper establishes a uniform procedure to split the operations in any algebraic operad, generalizing previous known notions of splitting algebraic structures from the dendriform algebra of Loday that splits the associative operation to…
Unlike the classical polynomial case there has not been invented up to very recently a tool similar to the Bernstein-Bezier representation which would allow us to control the behavior of the exponential polynomials. The exponential analog…
In this paper, we establish a bialgebra theory for Reynolds Lie algebras. First we introduce the notion of a quadratic Reynolds Lie algebra and show that it induces an isomorphism from the adjoint representation to the coadjoint…
A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…
The classical and quantum oscillator model on Lie-algebraically deformed nonrelativistic space-time is introduced and analyzed. The corresponding equations of motions are studied using mostly numerical methods. The time-dependent energy…
Using the language of operated algebras, we construct and investigate a class of operator rings and enriched modules induced by a derivation or Rota-Baxter operator. In applying the general framework to univariate polynomials, one is led to…
The main goal of this paper is to give some construction results of BiHom-post-Lie algebras which are a generalization of both post-Lie-algebras and Hom-post-Lie algebras. They are the algebraic structures behind the $\mathcal{O}$-operator…
Unifying various generalizations of the important notions of Reynolds operators, the relative cocycle weighted Reynolds operators are studied. Here cocycle weighted means the weight of the operators is given by a 2-cocycle rather than by a…