Related papers: Short Proof of Jacobi's Identity for Poisson Brack…
We introduce two remarkable identities written in terms of single commutators and anticommutators for any three elements of arbitrary associative algebra. One is a consequence of other (fundamental identity). From the fundamental identity,…
Capelli's and Turnbull's classical identities are given elegant combinatorial proofs.
For the standard symplectic forms on Jacobi and CMV matrices, we compute Poisson brackets of OPRL and OPUC, and relate these to other basic Poisson brackets and to Jacobians of basic changes of variable.
In this article, a short combinatorial proof of the Capelli's identity is given. It also leads to an easy proof of the Capelli--Cauchy--Binet identity, a more general form of Capelli's identity. With the technique introduced, the Turnbull's…
In this note, using the derangement polynomials and their umbral representation, we give another simple proof of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory.
We develop a canonical pairing between trees and graphs, which passes to their quotients by Jacobi identities. This pairing is an effective and simple tool for understanding the Lie and Poisson operads, providing canonical duals. In the…
In this paper, we introduce the notion of Jacobi polynomials with multiple reference vectors of a code, and give the MacWilliams type identity for it. Moreover, we derive a formula to obtain the Jacobi polynomials using the Aronhold…
We introduce the Poisson bracket operator which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in…
We show that several terminating summation and transformation formulas for basic hypergeometric series can be proved in a straightforward way. Along the same line, new finite forms of Jacobi's triple product identity and Watson's quintuple…
In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian…
We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi's canonical form of the hypergeometric differential equation. Analogy for $q$-hypergeometric…
We review the linearization of Poisson brackets and related problems, in the formal, analytic and smooth categories.
In this paper, we give the Jacobi identity of Toda brackets indexed by $n$, ($n \geq 0$), this generalizes Toda's Jacobi identity of Toda brackets indexed by 0 shown in [T].
In this paper we give an elementary proof of the Zariski-Lipman conjecture for log canonical spaces.
Easy steps through essential Hamiltonian dynamics are outlined, from necessary definitions of Poisson brackets and canonical transformations, to a quick proof that Hamiltonian evolution is made by canonical transformations, the quickest…
We give a simple and a more explicit proof of a mod $4$ congruence for a series involving the little $q$-Jacobi polynomials which arose in a recent study of a certain restricted overpartition function.
Cubical type theory provides a constructive justification of homotopy type theory. A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several…
The widely accepted approach to the foundation of quantum mechanics is that the Poisson bracket, governing the non-commutative algebra of operators, is taken as a postulate with no underlying physics. In this manuscript, it is shown that…
In this paper we present an equivalent statement to the Jacobian conjecture. For a polynomial map F on an affine space of dimension n, we define recursively n finite sequences of polynomials. We give an equivalent condition to the…
Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among…