In Ramanujan’s final letter to Hardy, he wrote of a strange new class of infinite series he called “mock theta functions”. It turns out all of Ramanujan’s mock theta functions are essentially specializations of a so-called universal mock theta function g3(z, q) of Gordon–McIntosh. Here we show that g3 arises naturally from the reciprocal of the classical Jacobi triple product—and is intimately ...

o bjectives: the purpose of this study was to evaluate the effect of salivary ph on the shear bond strength (sbs) of orthodontic brackets to tooth surface. materials and methods: eighty intact premolars were randomly divided into four groups of 20.  after bonding a bracket on each tooth, the groups one to four were stored in artificial saliva at a ph of 3.8, 4.8, 5.8, and 6.8, respectively for ...

The purpose of this study was to analyze the relationship between the critical contact angle and the torque angle in an orthodontic bracket and archwire assembly in 3 dimensions. Three-dimensional mathematical models were created with geometric bracket-archwire parameters that included 2 slot sizes, 3 bracket widths, and 3 to 4 wire sizes. From this, 3-dimensional mathematical equations (3DMEs)...

We construct the noncanonical Poisson bracket associated with the phase space of first order moments of the velocity field and quadratic moments of the density of a fluid with a free-boundary, constrained by the condition of incompressibility. Two methods are used to obtain the bracket, both based on Dirac’s procedure for incorporating constraints. First, the Poisson bracket of moments of the u...

We extend the Nambu bracket to 1-forms. Following the Poisson-Lie case, we define Nambu-Lie groups as Lie groups endowed with a multiplica-tive Nambu structure. A Lie group G with a Nambu structure P is a Nambu-Lie group iff P = 0 at the unit, and the Nambu bracket of left (right) invariant forms is left (right) invariant. We define a corresponding notion of a Nambu-Lie algebra. We give several...

The dynamics of Vlasov kinetic moments is shown to be Lie-Poisson on the dual Lie algebra of symmetric contravariant tensor fields. The corresponding Lie bracket is identified with the symmetric Schouten bracket and the moment Lie algebra is related with a bundle of bosonic Fock spaces, where creation and annihilation operators are used to construct the cold plasma closure. Kinetic moments are ...

The double bracket dissipation approach is applied to the Vlasov kinetic equation. The Vlasov equation is then transformed by a Poisson map to moment dynamics, leading to a nonlocal form of Darcy’s law. Next, kinetic equations for particles with anisotropic interaction are considered and also cast into the double bracket dissipation form. The moment dynamics for these double bracket kinetic equ...

When X is an associative H-space, the bar spectral sequence computes the homology of the delooping, H∗(BX). If X is an n-fold loop space for n ≥ 2 this is a spectral sequence of Hopf algebras. Using machinery by Sugawara and Clark, we show that the spectral sequence filtration respects the Browder bracket structure on H∗(BX), and so it is moreover a spectral sequence of Poisson algebras. Throug...

Relying on the notion of set-valued Lie bracket introduced in an earlier paper, we extend some classical results valid for smooth vector fields to the case when the vector fields are just Lipschitz. In particular, we prove that the flows of two Lipschitz vector fields commute for small times if and only if their Lie bracket vanishes everywhere (i.e., equivalently, if their classical Lie bracket...