We consider the nonlinear matrix equation X = Q+A (I⊗X−C)A (0 < δ ≤ 1), where Q is an n× n positive definite matrix, C is an mn×mn positive semidefinite matrix, I is the m×m identity matrix, and A is an arbitrary mn×n matrix. We prove the existence and uniqueness of the solution which is contained in some subset of the positive definite matrices under the condition that I ⊗Q > C. Two bounds for...

This problem generalizes the spectral norm of a matrix (p = q = 2) and the Grothendieck problem (p = ∞, q = 1), and has been widely studied in various regimes. When p ≥ q, the problem exhibits a dichotomy: constant factor approximation algorithms are known if 2 ∈ [q, p], and the problem is hard to approximate within almost polynomial factors when 2 / ∈ [q, p]. The regime when p < q, known as hy...

In recent years, substantial improvements were obtained in the effectiveness of data driven algorithms to validate the mapping of items to skills, or the Q-matrix. In the current study we use ensemble algorithms on top of existing Qmatrix refinement algorithms to improve their performance. We combine the boosting technique with a decision tree. The results show that the improvements from both t...

We consider the model of random surfaces with extrinsic curvature term embedded into 3d Euclidean lattice Z 3. On a 3d Euclidean lattice it has equivalent representation in terms of transfer matrix K(Q i , Q f), which describes the propagation of loops Q. We study the spectrum of the transfer matrix K(Q i , Q f) on finite dimensional lattices. The renormalisation group technique is used to inve...

We investigate small weight code words of the $p$-ary linear $\mathcal C_{j,k}(n,q)$ generated by incidence matrix $k$-spaces and $j$-spaces PG$(n,q)$ its dual, with $q$ a prime power $0 \leq j < k n$. Firstly, we prove that all up to $\left(3 - \mathcal{O}\left(\frac 1 q \right) \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q$ are combinations at most two (i.e. rows matrix). As for dual C_{j,k}(n,q)^\perp$,...