We show that near a point where the equilibrium density of eigenvalues of a matrix model behaves like y ∼ x p/q , the correlation functions of a random matrix, are, to leading order in the appropriate scaling, given by determinants of the universal (p, q)-minimal models kernels. Those (p, q) kernels are written in terms of functions solutions of a linear equation of order q, with polynomial coe...

Let p′ and q′ be points in R. Write p′ ∼ q′ if p′ − q′ is a multiple of (1, . . . , 1). Two different points p and q in R/ ∼ uniquely determine a tropical line L(p, q) passing through them and stable under small perturbations. This line is a balanced unrooted semi–labeled tree on n leaves. It is also a metric graph. If some representatives p′ and q′ of p and q are the first and second columns o...

We propose relevance determination and minimisation schemes in reinforcement learning which are solely based on the Q-matrix and which can thus be applied during training without prior knowledge about the system dynamics. On the one hand, we judge the relevance of separate state space dimensions based on the variance in the Q-matrix. On the other hand, we perform Q-matrix reduction by means of ...

This paper presents an algorithm to average a set of quaternion observations. The average quaternion is determined by minimizing the weighted sum of the squared Frobenius norms of the corresponding attitude matrix differences, subject to the unit-norm constraint in the determined solution. Two cases are presented: one that incorporates scalar weights and one that incorporates general weights on...

Let n n complex matrices P andQ be nontrivial generalized reflection matrices, i.e., P D P D P 1 ¤ In, Q DQ DQ 1 ¤ In. A complex matrix A with order n is said to be a .P;Q/ generalized anti-reflexive matrix, if PAQ D A. We in this paper mainly investigate the .P;Q/ generalized anti-reflexive maximal and minimal rank solutions to the system of matrix equation AX D B . We present necessary and su...

We present a condition on the matrix of an underdetermined linear system which guarantees that the solution of the system with minimal q-quasinorm is also the sparsest one. This generalizes, and slightly improves, a similar result for the 1norm. We then introduce a simple numerical scheme to compute solutions with minimal q-quasinorm, and we study its convergence. Finally, we display the result...

We use elements in the quantum hyperalgebra to define a quantum version of the Désarménien matrix. We prove that our matrix is upper triangular with ones on the diagonal and that, as in the classical case, it gives a quantum straightening algorithm for quantum bideterminants. We use our matrix to give a new proof of the standard basis theorem for the q-Weyl module. As well, we show that the sta...

then σ is a singular value of A and u and v are corresponding left and right singular vectors, respectively. (For generality it is assumed that the matrices here are complex, although given these results, the analogs for real matrices are obvious.) If, for a given positive singular value, there are exactly t linearly independent corresponding right singular vectors and t linearly independent co...

Let q be an odd natural number. We prove there is a cocyclic Hadamard matrix of order 210+tq whenever t ≥ 8b log2(q−1) 10 c. We also show that if the binary expansion of q contains N ones, then there is a cocyclic Hadamard matrix of order 24N−2q.

Using the notion of quantum integers associated with a complex number q 6= 0, we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little q-Jacobi polynomials when |q| < 1, and for the special value q = (1 − √ 5)/(1 + √ 5) they are closely related to Hankel matrices of reciprocal Fibonacci numbers called Filbert matrices. We find a formu...