arXiv subjects
Publications and source records attributed to Gabriel Pietrzkowski.
We consider a complete filtered Rota-Baxter algebra of weight $λ$ over a commutative ring. Finding the unique solution of a non-homogeneous linear algebraic equation in this algebra, we generalize Spitzer's identity in both commutative and non-commutative cases. As an application, considering the Rota-Baxter algebra of power series in one variable with q-integral as the Rota-Baxter operator, we show certain Eulerian identities.
We give a recursive formula for an expansion of a solution of a general non-autonomous polynomial differential equation. The formula is given on the algebraic level with a use of shuffle product. This approach minimizes the number of integrations on each order of expansion. Using combinatorics of trees we estimate the radius of convergence of the expansion.
For a general differential system $\dot x(t) = \sum_{d=1}^3 u_d(t)X_d$, where $X_d$ generates the simple Lie algebra of type $\mathfrak{a}_1$, we compute the explicit solution in terms of iterated integrals of products of $u_d$'s. As a byproduct we obtain the solution of a general Riccati equation by infinite quadratures.
We consider the problem of separability: decide whether a Hermitian operator on a finite dimensional Hilbert tensor product is separable or entangled. We show that the tensor convolution defined for certain mappings on an almost arbitrary locally compact abelian group, give rise to formulation of an equivalent problem to the separability one.
We study a separability problem suggested by mathematical description of bipartite quantum systems. We consider Hermitian 2-forms on the tensor product $H=K\otimes L$, where $K,L$ are finite dimensional complex spaces. Inspired by quantum mechanical terminology we call such a form separable if it is a convex combination of hermitian tensor products $(σ_p)^*\odot σ_p$ of 1-forms $σ_p$ on $H$ that are product forms $σ_p=ϕ_p\otimes ψ_p$, where $ϕ_p\in K^*$, $ψ_p\in L^*$. We introduce an integral representation of separable forms. In particular, we show that the integral of $(D_{z^*}}Φ)^*\odot D_{z^*}Φ$ of any square integrable map $Φ:\C^n\to \C^m$, with square integrable conjugate derivative $D_{z^*}Φ$, is a separable form. Vice versa, any separable form in the interior of the set of such forms, can be represented in this way. This implies that any separable mixed state (and only such states) can be either explicitly represented in the integral form, or it may be arbitrarily well approximated by such states.