We analyze necessary and sufficient conditions on a nonsingular matrix A such that, for any initial vector r₀, an orthogonal basis of the Krylov subspaces ${\cal K}_{n}(A,r_{0})$ is generated by a ...

Suppose we have a subspace \(\mathbb{S}\) of \(\mathbb{R}^n\) whose basis consists of \(k\) vectors \(\vec{v}_1,\vec{v}_2, \ldots , \vec{v}_k\). \[ \mathbb{S ...

Orthogonal polynomials have long played a pivotal role in quantum mechanics, offering a robust analytic framework to address exactly solvable models. Their applications range from providing explicit ...

We study what we call "dyadic CART"--a method of nonparametric regression which constructs a recursive partition by optimizing a complexity penalized sum of squares, where the optimization is over all ...

From RNA interference (RNAi) to CRISPR, there are several methods that researchers can use to manipulate gene function, each with its own strengths and weaknesses. Orthogonal validation – the ...