Leibniz Rule Research Articles

Odd Jacobi Manifolds and Loday-Poisson Brackets

We construct a nonskew symmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure and the Loday-Poisson structure. Furthermore, we show that the Loday-Poisson bracket satisfies the Leibniz rule over the noncommutative product derived from the homological vector field.

New Expansion Formulas for a Family of theλ-Generalized Hurwitz-Lerch Zeta Functions

We derive several new expansion formulas for a new family of theλ-generalized Hurwitz-Lerch zeta functions which were introduced by Srivastava (2014). These expansion formulas are obtained by making use of some important fractional calculus theorems such as the generalized Leibniz rules, the Taylor-like expansions in terms of different functions, and the generalized chain rule. Several (known or new) special cases are also considered.

A Generalization of the Leibniz Rule

We present a generalization of the Leibniz Rule. One of its consequences is the celebrated Abel identity.

Generalized derivations and general relativity

We construct differential geometry (connection, curvature, etc.) based on generalized derivations of an algebra [Formula: see text]. Such a derivation, introduced by Brešar in 1991, is given by a linear mapping [Formula: see text] such that there exists a usual derivation, d, of [Formula: see text] satisfying the generalized Leibniz rule u(ab) = u(a)b + ad(b) for all [Formula: see text]. The generalized geometry “is tested” in the case of the algebra of smooth functions on a manifold. We then apply this machinery to study generalized general relativity. We define the Einstein–Hilbert action and deduce from it Einstein’s field equations. We show that for a special class of metrics containing, besides the usual metric components, only one nonzero term, the action reduces to the O’Hanlon action that is the Brans–Dicke action with potential and with the parameter ω equal to zero. We also show that the generalized Einstein equations (with zero energy–stress tensor) are equivalent to those of the Kaluza–Klein theory satisfying a “modified cylinder condition” and having a noncompact extra dimension. This opens a possibility to consider Kaluza–Klein models with a noncompact extra dimension that remains invisible for a macroscopic observer. In our approach, this extra dimension is not an additional physical space–time dimension but appears because of the generalization of the derivation concept.

The Leibniz Rule for Fractional Derivatives Holds with Non-Differentiable Functions

In order to convince the sceptical reader, we herein give another proof of the fact that the Leibniz rule ) ( ) ( ) ( v D u v u D uv D α α α + =

Pointwise properties of functions of bounded variation in metric spaces

We study pointwise properties of functions of bounded variation on a metric space equipped with a doubling measure and a Poincare inequality. In particular, we obtain a Lebesgue type result for \(BV\) functions. We also study approximations by Lipschitz continuous functions and a version of the Leibniz rule. We give examples which show that our main result is optimal for \(BV\) functions in this generality.

An operator equation characterizing the Laplacian

The Laplace operator on Rn satisfies the equation ∆(fg)(x) = (∆f)(x)g(x) + f(x)(∆g)(x) + 2 〈f ′(x), g′(x)〉 for all f, g ∈ C2(Rn,R) and x ∈ Rn. We consider an operator equation generalizing this product formula. Suppose T : C2(Rn,R) → C(Rn,R) and A : C2(Rn,R) → C(Rn,Rn) are operators satisfying the equation T (fg)(x) = (Tf)(x)g(x) + f(x)(Tg)(x) + 〈(Af)(x), (Ag)(x)〉 (1) for all f, g ∈ C2(Rn,R) and x ∈ Rn . Assume, in addition, that T is O(n)-invariant, annihilates the affine functions and that A is non-degenerate. Then T is a multiple of the Laplacian on Rn , and A a multiple of the derivative, (Tf)(x) = d(||x||)2 2 (∆f)(x) , (Af)(x) = d(||x||)f ′(x) where d ∈ C(R+,R) is a continuous function. We describe the solutions as well, if T is not O(n)-invariant or does not annihilate the affine functions. To do so, we determine all operators (T,A) satisfying (1) for scalar operators A : C2(Rn,R) → C(Rn,R) . The map A, both in the vector and the scalar case, is closely related to T and there are just three different types of solution operators (T,A) . No continuity or linearity requirement is imposed on T or A . 1 Statement of the main results. It recently became clear that some fundamental operations in analysis and geometry like derivatives or duality maps are characterized by simple properties or operator functional equations: the derivative is characterized by the chain or the Leibniz rule, cf. [AAM], [AKM], [KM1], [KM2], the duality of convex bodies and the Legendre transform as a ∗Supported in part by the Alexander von Humboldt Foundation, by ISF grant 387/09 and BSF grant 2006079.

Universal Poisson Envelope for Binary-Lie Algebras

In this article the universal Poisson enveloping algebra for a binary-Lie algebra is constructed. Taking a basis 𝔹 of a binary-Lie algebra B, we consider the symmetric algebra S(B) of polynomials in the elements of 𝔹. We consider two products in S(B), the usual product of polynomials fg and the braces {f, g}, defined by the product in B and the Leibniz rule. This algebra is a general Poisson algebra. We find an ideal I of S(B) such that the factor algebra S(B)/I is the universal Poisson envelope of B. We provide some examples of this construction for known binary-Lie algebras.

DERIVATIVE INVERSE SERIES RELATIONS AND LAGRANGE EXPANSION FORMULA

A new pair of inverse series relations is established with the connection coefficients being expressed as higher derivatives of two fixed analytic functions. As applications, new proofs for the Lagrange expansion formulae are presented and Pfaff–Cauchy's generalizations of the Leibniz rule are reviewed.

A criterion for lattice supersymmetry: cyclic Leibniz rule

It is old folklore that the violation of Leibniz rule on a lattice is an obstruction for constructing a lattice supersymmetric model. While it is still true for full supersymmetry, we show that a slightly modified form of the Leibniz rule, which we call cyclic Leibniz rule (CLR), is actually a criterion for the existence of partial lattice supersymmetry. In one dimension, we find sets of lattice difference operator and field multiplication smeared over lattice which satisfy the CLR under some natural assumptions such as translational invariance and locality. Thereby we construct a model of supersymmetric lattice quantum mechanics without spoiling locality. The CLR relation is coincident with the condition that the vanishing of the so-called surface term in the construction by lattice Nicolai map. We can construct superfield formalism with arbitrary superpotential. This also enables us to apply safely a localization technique to our model, because the kinetic term and the interaction terms of our model are independently invariant under the supersymmetry transformation. A preliminary attempt in finding a solution for the higher dimensional case is also discussed.

No violation of the Leibniz rule. No fractional derivative

We demonstrate that a violation of the Leibniz rule is a characteristic property of derivatives of non-integer orders. We prove that all fractional derivatives Dα, which satisfy the Leibniz rule Dα(fg)=(Dαf)g+f(Dαg), should have the integer order α=1, i.e. fractional derivatives of non-integer orders cannot satisfy the Leibniz rule.

Some relations involving a generalized fractional derivative operator

Recently, Katugampola (Appl. Math. Comput. 218:860-865, 2011) studied a special case of the Erdélyi-Kober generalized fractional derivative. This special case generalized the well-known Riemann-Liouville and the Hadamard fractional integrals into a single form. Katugampola denoted this special case by the operator D x α 0 ρ . Some properties and examples for this fractional derivative operator was given. In this paper, we present some additional properties for this operator defined, this time, in the complex plane. In particular, we express this fractional derivative operator in terms of the classical Riemann-Liouville fractional derivative operator. A generalized Leibniz rule is obtained.MSC:26A33, 33C45.

Rigidity and stability of the Leibniz and the chain rule

We study rigidity and stability properties of the Leibniz and chain rule operator equations. We describe which non-degenerate operators V, T1, T2,A: Ck(ℝ) → C(ℝ) satisfy equations of the generalized Leibniz and chain rule type for f, g ∈ Ck(ℝ), namely, V (f · g) = (T1f) · g + f · (T2g) for k = 1, V (f · g) = (T1f) · g + f · (T2g) + (Af) · (Ag) for k = 2, and V (f ○ g) = (T1f) ○ g · (T2g) for k = 1. Moreover, for multiplicative maps A, we consider a more general version of the first equation, V (f · g) = (T1f) · (Ag) + (Af) · (T2g) for k = 1. In all these cases, we completely determine all solutions. It turns out that, in any of the equations, the operators V, T1 and T2 must be essentially equal. We also consider perturbations of the chain and the Leibniz rule, T (f ○ g) = Tf ○ g · Tg + B(f ○ g, g) and T (f · g) = Tf · g + f · Tg + B(f, g), and show under suitable conditions on B in the first case that B = 0 and in the second case that the solution is a perturbation of the solution of the standard Leibniz rule equation.

Braided algebras and their applications to Noncommutative Geometry

We introduce the notion of a braided algebra and study some examples of these. In particular, R-symmetric and R-skew-symmetric algebras of a linear space V equipped with a skew-invertible Hecke symmetry R are braided algebras. We prove the “mountain property” for the numerators and denominators of their Poincaré–Hilbert series (which are always rational functions).Also, we further develop a differential calculus on modified Reflection Equation algebras. Thus, we exhibit a new form of the Leibniz rule for partial derivatives on such algebras related to involutive symmetries R. In particular, we present this rule for the algebra U(gl(m)). The case of the algebra U(gl(2)) and its compact form U(u(2)) (which can be treated as a deformation of the Minkowski space algebra) is considered in detail. On the algebra U(u(2)) we introduce the notion of the quantum radius, which is a deformation of the usual radius, and compute the action of rotationally invariant operators and in particular of the Laplace operator. This enables us to define analogs of the Laplace–Beltrami operators corresponding to certain Schwarzschild-type metrics and to compute their actions on the algebra U(u(2)) and its central extension. Some “physical” consequences of our considerations are presented.

Water wave propagation equation from expanded form of Leibniz rule

The Leibniz integral rule for the first derivative with variable limits of integration is expanded for the second and the third order derivatives by using the chain rule here. Water wave propagation equation in the x-z plane containing second derivatives is depth-integrated by using the Leibniz rule. The integrated wave equation is then applied to wave transmission and reflection over an inclined step, and the computed reflection coefficients agree well with those from existing theories such as full linear equation, mild slope equation, and modified mild slope equation.

A new Leibniz rule and its integral analogue for fractional derivatives

In 1972, T.J. Osler proposed a generalization of the Leibniz rule for the fractional derivatives of the product of two functions with respect to an arbitrary function. This new rule was based on one of his own result on Taylor's series for the fractional derivative he obtained in 1971. Later, he gave the integral analogue of that new Leibniz rule. In this paper, we present a new Leibniz rule for the fractional derivatives of the product of two functions with respect to an arbitrary function and we give its integral analogue. Finally, new series expansions and definite integrals involving special functions are derived as special cases of the new Leibniz rule and the corresponding integral analogue.

Duality on the quantum space(3) with two parameters

Abstract In this study, we introduce a dual Hopf algebra in the sense of Sudbery for the quantum space(3) whose coordinates satisfy the commutation relations with two parameters and we show that the dual algebra is isomorphic to the quantum Lie algebra corresponding to the Cartan-Maurer right invariant differential forms on the quantum space(3). We also observe that the quantum Lie algebra generators are commutative as those of the undeformed Lie algebra and the deformation becomes apparent when one studies the Leibniz rules for the generators.

REPRESENTATIONS OF sl(2) IN THE BOOLEAN LATTICE, AND THE HAMMING AND JOHNSON SCHEMES

Starting with the zero-square "zeon algebra," the regular representation gives rise to a Boolean lattice representation of sl(2). We detail the su(2) content of the Boolean lattice, providing the irreducible representations carried by the algebra generated by the subsets of an n-set. The group elements are found, exhibiting the "special functions" in this context. The corresponding Leibniz rule and group law are shown. Krawtchouk polynomials, the Hamming and the Johnson schemes appear naturally. Applications to the Boolean poset and the structure of Hadamard–Sylvester matrices are shown as well.

Hardy type derivations on generalised series fields

We consider the valued field K : = R ( ( Γ ) ) of generalised series (with real coefficients and monomials in a totally ordered multiplicative group Γ). We investigate how to endow K with a series derivation, that is a derivation that satisfies some natural properties such as commuting with infinite sums (strong linearity) and (an infinite version of) Leibniz rule. We characterise when such a derivation is of Hardy type, that is, when it behaves like differentiation of germs of real valued functions in a Hardy field. We provide a necessary and sufficient condition for a series derivation of Hardy type to be surjective.

CALCULUS ON FRACTAL SUBSETS OF REAL LINE — II: CONJUGACY WITH ORDINARY CALCULUS

Calculus on fractals, or Fα-calculus, developed in a previous paper, is a calculus based fractals F ⊂ R, and involves Fα-integral and Fα-derivative of orders α, 0 < α ≤ 1, where α is the dimension of F. The Fα-integral is suitable for integrating functions with fractal support of dimension α, while the Fα-derivative enables us to differentiate functions like the Cantor staircase. Several results in Fα-calculus are analogous to corresponding results in ordinary calculus, such as the Leibniz rule, fundamental theorems, etc. The functions like the Cantor staircase function occur naturally as solutions of Fα-differential equations. Hence the latter can be used to model processes involving fractal space or time, which in particular include a class of dynamical systems exhibiting sublinear behaviour. In this paper we show that, as operators, the Fα-integral and Fα-derivative are conjugate to the Riemann integral and ordinary derivative respectively. This is accomplished by constructing a map ψ which takes Fα-integrable functions to Riemann integrable functions, such that the corresponding integrals on appropriate intervals have equal values. Under suitable conditions, a restriction of ψ also takes Fα-differentiable functions to ordinarily differentiable functions such that their values at appropriate points are equal. Further, this conjugacy is generalized to one between Sobolev spaces in ordinary calculus and Fα-calculus. This conjugacy is useful, among other things, to find solutions to Fα-differential equations: they can be mapped to ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a few examples.