Remarks on Robin's and Niocolas Inequalities
Robin's Conjecture is strengthened, deformed, and proved. Nicolas conjecture follows.
math-ph↗
arXiv subjects
Boris A. Kupershmidt
Publications and source records attributed to Boris A. Kupershmidt.
KdV6: An Integrable System
$K^2 S^2 T [5]$ recently derived a new 6$^{th}$-order wave equation $KdV6$: $(\partial^2_x + 8u_x \partial_x + 4u_{xx})(u_t + u_{xxx} + 6u_x^2) = 0$, found a linear problem and an auto-B${\ddot{\rm{a}}}$ckclund transformation for it, and conjectured its integrability in the usual sense. We prove this conjecture by constructing an infinite commuting hierarchy $KdV_n6$ with a common infinite set of conserved densities. A general construction is presented applicable to any bi-Hamiltonian system (such as all standard Lax equations, continuous and discrete) providing a nonholonomic perturbation of it. This perturbation is conjectured to preserve integrability. That conjecture is verified in a few representative cases: the classical long-wave equations, the Toda lattice (both continuous and discrete), and the Euler top.
nlin.SI↗
Dark equations
Observing the Universe, astronomers have concluded that the motion of stars can not be accounted for unless one assumes that most of the mass in the Universe is carried on by a ``dark matter", so far impervious to all attempts at being detected. There is now a similar concept of ``dark energy". I shall discuss a different subject, ``dark equations". These have never indicated that they influence anything or even exist, but if one supposes that they do exist, one can systematically discover them and study their properties, some of which turn out to be strange and others mysterious. These equations are similar in spirit to what one gets when linearizing a given system, or studies how an external linear wave interacts with a particular solution of a given system. We define and study linear extensions of dynamical systems in general, and integrable and Hamiltonian systems in particular. Systems discussed include the KdV and mKdV equations and the associated Miura maps, the Burgers hierarchy and the associated Hopf--Cole transformations, long wave equations, the Benney hierarchy, and the KP hierarchy.
nlin.SI↗
And free lunch for all...
This is a review of the 5-volumes of Ramanujan's Notebooks, as worked over by Bruce C. Berndt over the last quarter of the XX-th Century. To illustrate how useful Ramanujan's insights could be for anyone who indulges in the wild pleasure of savoring Ramanujan's genius, the reviewer takes one formula of Ramanujan from the Notebooks, analyses it carefully, and arrives at the beginnings of what could be fairly called "Quantum Number Theory."
math.HO↗
q-Newton binomial: from Euler to Gauss
A counter-intuitive result of Gauss (formulae (1.6), (1.7) below) is made less mysterious by virtue of being generalized through the introduction of an additional parameter.
math.HO↗
q-probability: I. Basic discrete distributions
For basic discrete probability distributions, $-$ Bernoulli, Pascal, Poisson, hypergeometric, contagious, and uniform, $-$ $q$-analogs are proposed.
math.PR↗
What a classical r-matrix really is
The notion of classical $r$-matrix is re-examined, and a definition suitable to differential (-difference) Lie algebras, -- where the standard definitions are shown to be deficient, -- is proposed, the notion of an ${\mathcal O}$-operator. This notion has all the natural properties one would expect form it, but lacks those which are artifacts of finite-dimensional isomorpisms such as not true in differential generality relation $\mbox{End}\, (V) \simeq V^* \otimes V$ for a vector space $V$. Examples considered include a quadratic Poisson bracket on the dual space to a Lie algebra; generalized symplectic-quadratic models of such brackets (aka Clebsch representations); and Drinfel'd's 2-cocycle interpretation of nondegenate classical $r$-matrices.
math.QA↗
Remarks on quantization of classical r-matrices
If a classical $r$-matrix $r$ is skewsymmetric, its quantization $R$ can lose the skewsymmetry property. Even when $R$ is skewsymmetric, it may not be unique.
math.QA↗
Coadjoint Poisson actions of Poisson-Lie groups
A Poisson-Lie group acting by the coadjoint action on the dual of its Lie algebra induces on it a non-trivial class of quadratic Poisson structures extending the linear Poisson bracket on the coadjoint orbits.
math.QA↗
On the nature of the Virasoro algebra
The multiplication in the Virasoro algebra \[ [e_p, e_q] = (p - q) e_{p+q} + θ\left(p^3 - p\right) δ_{p + q}, \qquad p, q \in {\mathbf Z}, \] \[ [θ, e_p] = 0, \] comes from the commutator $[e_p, e_q] = e_p * e_q - e_q * e_p$ in a quasiassociative algebra with the multiplication \renewcommand{\theequation}{$*$} \be \ba{l} \ds e_p * e_q = - {q (1 + εq) \over 1 + ε(p + q)} e_{p+q} + {1 \over 2} θ\left[p^3 - p + \left(ε- ε^{-1} \right) p^2 \right] δ^0_{p+q}, \vspace{3mm}\\ \ds e_p * θ= θ* e_p = 0. \ea \ee The multiplication in a quasiassociative algebra ${\cal R}$ satisfies the property \renewcommand{\theequation}{$**$} \be a * (b * c) - (a * b) * c = b * (a * c) - (b * a) * c, \qquad a, b, c \in {\cal R}. \ee This property is necessary and sufficient for the Lie algebra {\it Lie}$({\cal R})$ to have a phase space. The above formulae are put into a cohomological framework, with the relevant complex being different from the Hochschild one even when the relevant quasiassociative algebra ${\cal R}$ becomes associative. Formula $(*)$ above also has a differential-variational counterpart.
math.QA↗
Remarks on random evolutions in Hamiltonian representation
telegrapher's equations and some random walks of Poisson type are shown to fit into the framework of the Hamiltonian formalism after an appropriate time-dependent rescaling of the basic variables has been made.
math-ph↗
Quantum differential forms
Formalism of differential forms is developed for a variety of Quantum and noncommutative situations.
quant-ph↗
Hamiltonian Formalism in Quantum Mechanics
Heisenberg motion equations in Quantum mechanics can be put into the Hamilton form. The difference between the commutator and its principal part, the Poisson bracket, can be accounted for exactly. Canonical transformations in Quantum mechanics are not, or at least not what they appear to be; their properties are formulated in a series of Conjectures.
quant-ph↗