Related papers: On algebraic Stein operators for Gaussian polynomi…
We study the problem of constructing universal Steiner trees for undirected graphs. Given a graph $G$ and a root node $r$, we seek a single spanning tree $T$ of minimum {\em stretch}, where the stretch of $T$ is defined to be the maximum…
Our main result is a full classification, for every connected graph $H$, of the computational complexity of Steiner Forest on $H$-subgraph-free graphs. To obtain this dichotomy, we establish the following new algorithmic, hardness, and…
We present other examples illustrating the operator-theoretic approach to invariant integrals on quantum homogeneous spaces developed by Kuersten and the second author. The quantum spaces are chosen such that their coordinate algebras do…
Operator-splitting methods are widely used to solve differential equations, especially those that arise from multi-scale or multi-physics models, because a monolithic (single-method) approach may be inefficient or even infeasible. The most…
Let $(W,S)$ be a finite Coxeter system with root system $R$ and with set of positive roots $R^+$. For $\alpha\in R$, $v,w\in W$, we denote by $\partial_\alpha$, $\partial_w$ and $\partial_{w/v}$ the divided difference operators and skew…
Let $X^N = (X_1^N,\dots, X^N_d)$ be a d-tuple of $N\times N$ independent GUE random matrices and $Z^{NM}$ be any family of deterministic matrices in $\mathbb{M}_N(\mathbb{C})\otimes \mathbb{M}_M(\mathbb{C})$. Let $P$ be a self-adjoint…
Estimating the score, i.e., the gradient of log density function, from a set of samples generated by an unknown distribution is a fundamental task in inference and learning of probabilistic models that involve flexible yet intractable…
We consider general Gaussian latent tree models in which the observed variables are not restricted to be leaves of the tree. Extending related recent work, we give a full semi-algebraic description of the set of covariance matrices of any…
Using the language of operated algebras, we construct and investigate a class of operator rings and enriched modules induced by a derivation or Rota-Baxter operator. In applying the general framework to univariate polynomials, one is led to…
Let $G_\C$ be a connected, linear algebraic group defined over $\R$, acting regularly on a finite dimensional vector space $V_\C$ over $\C$ with $\R$-structure $V_\R$. Assume that $V_\C$ posseses a Zariski-dense orbit, so that…
We show how, given a non-Hermitian Hamiltonian $H$, we can generate new non-Hermitian operators sequentially, producing a virtually infinite chain of non-Hermitian Hamiltonians which are isospectral to $H$ and $H^\dagger$ and whose…
We apply concepts of random differential geometry connected to the random matrix ensembles of the random linear operators acting on finite dimensional Hilbert spaces. The values taken by random linear operators belong to the Liouville…
A method of generating differential operators is used to solve the spectral problem for a generalisation of the Sylvester-Kac matrix. As a by-product, we find a linear differential operator with polynomial coefficients of the first order…
Stein's method for measuring convergence to a continuous target distribution relies on an operator characterizing the target and Stein factor bounds on the solutions of an associated differential equation. While such operators and bounds…
We introduce higher-order Stein kernels relative to the standard Gaussian measure, which generalize the usual Stein kernels by involving higher-order derivatives of test functions. We relate the associated discrepancies to various metrics…
We introduce Wirtinger operators for functions of several quaternionic variables. These operators are real linear partial differential operators which behave well on quaternionic polynomials, with properties analogous to the ones satisfied…
The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space $\mathcal{H}$ with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators…
We introduce the concept of $\D$-operators associated to a sequence of polynomials $(p_n)_n$ and an algebra $\A$ of operators acting in the linear space of polynomials. In this paper, we show that this concept is a powerful tool to generate…
We prove an analogue of the Central Limit Theorem for operators. For every operator $K$ defined on $\mathbb{C}[x]$ we construct a sequence of operators $K_N$ defined on $\mathbb{C}[x_1,...,x_N]$ and demonstrate that, under certain…
In this monograph we develop magnetic pseudodifferential theory for operator-valued and equivariant operator-valued functions and distributions from first principles. These have found plentiful applications in mathematical physics,…