Quadratic Poisson brackets and Drinfeld theory for associative algebras

The paper is devoted to the Poisson brackets compatible with multiplication in associative algebras. These brackets are shown to be quadratic and their relations with the classical Yang--Baxter equation are revealed. The paper also contains a description of Poisson Lie structures on Lie groups whose Lie algebras are adjacent to an associative structure.Comment: 16 pages, latex, no figure

On the algebraic structures connected with the linear Poisson brackets of hydrodynamics type

The generalized form of the Kac formula for Verma modules associated with linear brackets of hydrodynamics type is proposed. Second cohomology groups of the generalized Virasoro algebras are calculated. Connection of the central extensions with the problem of quntization of hydrodynamics brackets is demonstrated

Quadratic Poisson brackets and Drinfel'd theory for associative algebras

Quadratic Poisson brackets on associative algebras are studied. Such a bracket compatible with the multiplication is related to a differentiation in tensor square of the underlying algebra. Jacobi identity means that this differentiation satisfies a classical Yang--Baxter equation. Corresponding Lie groups are canonically equipped with a Poisson Lie structure. A way to quantize such structures is suggested.Comment: latex, no figures

Enhancing Conformal Prediction Using E-Test Statistics

Conformal Prediction (CP) serves as a robust framework that quantifies uncertainty in predictions made by Machine Learning (ML) models. Unlike traditional point predictors, CP generates statistically valid prediction regions, also known as prediction intervals, based on the assumption of data exchangeability. Typically, the construction of conformal predictions hinges on p-values. This paper, however, ventures down an alternative path, harnessing the power of e-test statistics to augment the efficacy of conformal predictions by introducing a BB-predictor (bounded from the below predictor)

On the virial theorem for the relativistic operator of Brown and Ravenhall, and the absence of embedded eigenvalues

A virial theorem is established for the operator proposed by Brown and Ravenhall as a model for relativistic one-electron atoms. As a consequence, it is proved that the operator has no eigenvalues greater than $\max(m c^2, 2 \alpha Z - \frac{1}{2})$, where $\alpha$ is the fine structure constant, for all values of the nuclear charge $Z$ below the critical value $Z_c$: in particular there are no eigenvalues embedded in the essential spectrum when $Z \leq 3/4 \alpha$. Implications for the operators in the partial wave decomposition are also described.Comment: To appear in Letters in Math. Physic

Dirac-Sobolev inequalities and estimates for the zero modes of massless Dirac operators

The paper analyses the decay of any zero modes that might exist for a massless Dirac operator H:= \ba \cdot (1/i) \bgrad + Q, where $Q$ is $4 \times 4$-matrix-valued and of order O(|\x|^{-1}) at infinity. The approach is based on inversion with respect to the unit sphere in $\R^3$ and establishing embedding theorems for Dirac-Sobolev spaces of spinors $f$ which are such that $f$ and $Hf$ lie in $(L^p(\R^3))^4, 1\le p<\infty.$Comment: 11 page