Search

In this review, we discuss currents in celestial CFT and the consistency of their naïve symmetry algebras. In particular, we study in detail the Jacobi identity and the double residue condition for soft insertions, hard momentum space insertions, and hard celestial insertions. In the latter case, we introduce the notion of a “hard current” in CFT and work through examples in the 2D critical Ising model. The current algebra of hard insertions in pure Einstein gravity is a slight conceptual generalization of the familiar w1+∞-wedge current algebra. We also review branch cut terms in the celestial OPE, which indicate new primary content and were previously missed until recently. We work through an explicit toy example illustrating the mechanism by which such branch cut terms can arise. These branch cut terms prevent a symmetry interpretation but are fully compatible with a consistent OPE. [ABSTRACT FROM AUTHOR]

Mock‐Lie algebras, a unique class of commutative algebras that satisfy the Jacobi identity, are gaining attention for their potential applications in various mathematical contexts. This study presents a matrix representation of the centroid for Mock‐Lie algebras in dimensions up to four, offering insights into their structure and interactions. By applying the centroid concept and utilizing matrix operations, we explore the relationships between matrix elements, leading to a better understanding of these algebras. Our findings aim to contribute to the growing interest in Mock‐Lie algebras and their role within the broader algebraic landscape. [ABSTRACT FROM AUTHOR]

Let G be a connected graph with V (G) = { 1 , ... , n }. Then the resistance distance between any two vertices i and j is given by r i j := l i i † + l j j † − 2 l i j † , where l i j † is the (i , j) th entry of the Moore-Penrose inverse of the Laplacian matrix of G. For the resistance matrix R := [ r i j ] , there is an elegant formula to compute the inverse of R. This says that R − 1 = − 1 2 L + 1 τ ′ R τ τ τ ′ , where τ := (τ 1 , ... , τ n) ′ and τ i := 2 − ∑ j ∼ i r i j i = 1 , ... , n. A far reaching generalization of this result that gives an inverse formula for a generalized resistance matrix of a strongly connected and matrix weighted balanced directed graph is obtained in this paper. When the weights are scalars, it is shown that the generalized resistance is a non-negative real number. We also obtain a perturbation result involving resistance matrices of connected graphs and Laplacians of digraphs. [ABSTRACT FROM AUTHOR]

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics. [ABSTRACT FROM AUTHOR]

A higher dimensional generalization of the cross product is associated with an adequate matrix multiplication. This index-free view allows for a better understanding of the underlying algebraic structures, among which are generalizations of Grassmann’s, Jacobi’s and Room’s identities. Moreover, such a view provides a higher dimensional analogue of the decomposition of the vector Laplacian, which itself gives an explicit index-free Helmholtz decomposition in arbitrary dimensions n≥2n\ge 2.

In this note, we give a short proof of a modular equation of Ramanujan–Selberg continued fraction given by Chan (Proc Am Math Soc 137:2849–2856, 2009). [ABSTRACT FROM AUTHOR]

Nambu dynamics is a generalized Hamiltonian dynamics of more than two variables, whose time evolutions are given by the Nambu bracket, a generalization of the canonical Poisson bracket. Nambu dynamics can always be represented in the form of noncanonical Hamiltonian dynamics by defining the noncanonical Poisson bracket by means of the Nambu bracket. For the time evolution to be consistent, the Nambu bracket must satisfy the fundamental identity, while the noncanonical Poisson bracket must satisfy the Jacobi identity. However, in many degrees of freedom systems, it is well known that the fundamental identity does not hold. In the present paper we show that, even if the fundamental identity is violated, the Jacobi identity for the corresponding noncanonical Hamiltonian dynamics could hold. As an example we evaluate these identities for a semiclassical system of two coupled oscillators. [ABSTRACT FROM AUTHOR]

Classical Hamiltonian mechanics is realized by the action of a Poisson bracket on a Hamiltonian function. The Hamiltonian function is a constant of motion (the energy) of the system. The properties of the Poisson bracket are encapsulated in the symplectic |$2$| -form, a closed second-order differential form. Due to closure, the symplectic |$2$| -form is preserved by the Hamiltonian flow, and it assigns an invariant (Liouville) measure on the phase space through the Lie–Darboux theorem. In this paper we propose a generalization of classical Hamiltonian mechanics to a three-dimensional phase space: the classical Poisson bracket is replaced with a generalized Poisson bracket acting on a pair of Hamiltonian functions, while the symplectic |$2$| -form is replaced by a symplectic |$3$| -form. We show that, using the closure of the symplectic |$3$| -form, a result analogous to the classical Lie–Darboux theorem holds: locally, there exist smooth coordinates such that the components of the symplectic |$3$| -form are constants, and the phase space is endowed with a preserved volume element. Furthermore, as in the classical theory, the Jacobi identity for the generalized Poisson bracket mathematically expresses the closure of the associated symplectic form. As a consequence, constant skew-symmetric third-order contravariant tensors always define generalized Poisson brackets. This is in contrast with generalizations of Hamiltonian mechanics postulating the fundamental identity as replacement for the Jacobi identity. In particular, we find that the fundamental identity represents a stronger requirement than the closure of the symplectic |$3$| -form. [ABSTRACT FROM AUTHOR]

Twisted Jacobi structures [24] are faced with Jacobi with background structures [4, 5]. The comparative analysis is done both for trivial and non-trivial line bundle versions. [ABSTRACT FROM AUTHOR]

In this paper, we find conditions under which the bracket defined by a graded derivation on a Lie superalgebra (g; [; ]) is skew-supersymmetry and satisfies the super Jacobi identity, so it defines the structure of a Lie superalgebra on g:. In the case of the algebra of differential forms on a supermanifold, we study the graded commutator of graded derivations, graded skew-derivations and a graded derivation, with another graded skew-derivation of the superalgebra of differential forms on a supermanifold. [ABSTRACT FROM AUTHOR]

The multiconfigurational time-dependent Hartree (MCTDH) approach facilitates multidimensional quantum dynamics calculations by employing a layered representation: in the upper layer of the representation the multidimensional wave function is expanded in a basis of time-dependent single-particle functions, in the lower layer then these single-particle functions are represented in a larger time-independent basis. This two layer approach can be generalized to a multilayer MCTDH representation to further increase the numerical efficiency. In recent work [U. Manthe, J. Chem. Phys. 128, 164116 (2008)], a quadrature scheme has been introduced which facilitates multilayer MCTDH calculations for Hamiltonians with general potential energy functions. Based on time-dependent discrete variable representation (DVR) grids corresponding to the single-particle function bases the scheme employs the correlation DVR (CDVR) approach. The present work shows that serious problems arise when this original multilayer CDVR approach is applied for accurate calculations of vibrational states. A solution to these problems is presented which uses a revised scheme to define the time-dependent DVR grids entering into the CDVR scheme. Applications studying the vibrational states of CH3 demonstrate that the resulting revised multilayer MCTDH/CDVR approach yields smoothly convergent and accurate results. [ABSTRACT FROM AUTHOR]

We show that the maximum cardinality of an equiangular line system in R 18 is at most 59. Our proof includes a novel application of the Jacobi identity for complementary subgraphs. In particular, we show that there does not exist a graph whose adjacency matrix has characteristic polynomial (x − 22) (x − 2) 42 (x + 6) 15 (x + 8) 2. [ABSTRACT FROM AUTHOR]

A quantum-classical bracket is proposed and is shown to satisfy the Jacobi identity, in contrast to previous definitions that obey this property only up to higher order terms in the Planck constant h. The Jacobi identity is required of a true Lie bracket and ensures that the Lie bracket of constants of motion is also a constant of motion. An explicit calculation of the Jacobi identity highlights the difference between the proposed and traditional definitions. A further example illustrates that the proposed bracket generates a more consistent quantum-classical dynamics than the traditional bracket. The traditional quantum-classical dynamics in the Henon-Heiles system diverges due to higher order h terms. The divergence is eliminated with the proposed bracket. [ABSTRACT FROM AUTHOR]

Abstract Based on the theory of Poisson vertex algebras we calculate skew-symmetry conditions and Jacobi identities for a class of third-order nonlocal operators of differential-geometric type. Hamiltonian operators within this class are defined by a Monge metric and a skew-symmetric two-form satisfying a number of differential-geometric constraints. Complete classification results in the 2-component and 3-component cases are obtained. [ABSTRACT FROM AUTHOR]

Abstract Multi-trace formulations (MTFs) are based on a decomposition of the problem domain into subdomains, and thus domain decomposition solvers are of interest. The fully rigorous mathematical MTF can however be daunting for the non-specialist. The first aim of the present contribution is to provide a gentle introduction to MTFs. We introduce these formulations on a simple model problem using concepts familiar to researchers in domain decomposition. This allows us to get a new understanding of MTFs and a natural block Jacobi iteration, for which we determine optimal relaxation parameters. We then show how iterative multi-trace formulation solvers are related to a well known domain decomposition method called optimal Schwarz method: a method which used Dirichlet to Neumann maps in the transmission condition. We finally show that the insight gained from the simple model problem leads to remarkable identities for Calderón projectors and related operators, and the convergence results and optimal choice of the relaxation parameter we obtained is independent of the geometry, the space dimension of the problem, and the precise form of the spatial elliptic operator, like for optimal Schwarz methods. We illustrate our analysis with numerical experiments. [ABSTRACT FROM AUTHOR]

We interpret the second difference of the sequence of the number of strict partitions, for n ≥ 5 , as the sequence of the strict partitions of n with at least three parts , the three largest parts consecutive , and the smallest part at least two . The name butterfly describes both a sequence's interpretation and a bijection between subsets of strict partitions. Using cyclotomic polynomials, we compute generating function identities of the butterfly sequence and related sequences both as infinite products and as series filtered by the number of parts. We offer a merging and splitting construction of the butterfly sequence as a sequence of partitions with odd parts larger or equal to 3, and we interpret the butterfly sequence as a sequence of generalized pentagonal , pentagonal with domino , and non-pentagonal butterfly partitions . Euler's Pentagonal Number Theorem and a similar specialization of the Jacobi Triple Product lead to recursive algorithms to compute the butterfly sequence and related sequences. [ABSTRACT FROM AUTHOR]

We verify the maximum conjecture on the rigidity of totally nondegenerate model CR manifolds in the following two cases: (i) for all models of CR dimension one, (ii) for the so-called full-models, namely those in which their associated symbol algebras are free CR. In particular, we discover that in each arbitrary CR dimension and length ≥ 3 {\geq 3} , there exists at least one totally nondegenerate model, enjoying this conjecture. Our proofs rely upon some recent results in the Tanaka theory of transitive prolongation of fundamental algebras. [ABSTRACT FROM AUTHOR]

We study several classes of non-associative algebras as possible candidates for deformation quantization in the direction of a Poisson bracket that does not satisfy Jacobi identities. We show that in fact alternative deformation quantization algebras require the Jacobi identities on the Poisson bracket and, under very general assumptions, are associative. At the same time, flexible deformation quantization algebras exist for any Poisson bracket. [ABSTRACT FROM AUTHOR]

Let L be a Filippov algebra. A derivation of L is called an ID -derivation if its image is contained in the derived algebra of L. Let ID ∗ ( L ) be the set of all ID -derivations which map central elements to 0. We prove that ID ∗ ( L ) and ID ∗ ( M ) are isomorphic for any two isoclinic Filippov algebras L and M. [ABSTRACT FROM AUTHOR]

In this paper, we show certain series identities arising from the Jacobi identity of the ordinary theta function. These include several formulas of Ramanujan type given by Berndt and their relevant analogues. [ABSTRACT FROM AUTHOR]

In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-algebras from A(m)-algebras, restricted Leibniz algebras, restricted right-symmetric algebras. Finally, we prove that there is a one-to-one correspondence between strict restricted Lie 2-algebras and crossed modules of restricted Lie algebras. [ABSTRACT FROM AUTHOR]

In this paper, we study third power associative, antiflexible rings satisfying the identity (a,b,ac) = a(a,b,c). We prove that third power associative, antiflexible rings satisfying the identity (a,b,ac) = a(a,b,c) with characteristic ≠2,3 are associative of degree 5. As a consequence of this result, we prove that a third power associative semiprime antiflexible ring satisfying the identity (a,b,ac) = a(a,b,c) is associative. [ABSTRACT FROM AUTHOR]

For α, β ∈ N0 and max{a, ß} > 0, it is shown that the integrals of the Jacobi polynomials t 0 (t - ...)dP (a-12, ß-12) n (cos ...) sin ... 2a cos ... 2ß d > 0 for all t ∈ (0, p] and n ∈ N if d = a+1 for a, ß ∈ N0 and max{a, ß} > 0. This proves a conjecture on the integral of the Gegenbauer polynomials in a work of Beatson (2014) that implies the strictly positive definiteness of the function θ → (t - θ) + on the unit sphere Sd-1 for d = d 2 ... and the P'olya criterion for positive definite functions on the sphere Sd-1 for d = 3. Moreover, the positive definiteness of the function ... + is also established on the compact two-point homogeneous spaces. [ABSTRACT FROM AUTHOR]

We consider matrices over a commutative ring and characterize the class of outer inverses for which Jacobi type identities can be extended. We obtain a necessary and sufficient condition for the existence of Rao-regular Drazin inverse in terms of sum of principal minors of A k for some k . Also, we obtain determinantal formula for the Rao-regular Drazin inverse. Conjectures are formulated which give expressions for outer inverse and the conjectures are proved in some special cases. [ABSTRACT FROM AUTHOR]

We develop a previously proposed algebraic technique for a Hamiltonian approach to evolution systems of partial differential equations including constrained systems and propose a defining system of equations ( suitable for computer calculations) characterizing the Hamiltonian operators of a given form. We demonstrate the technique with a simple example. [ABSTRACT FROM AUTHOR]