Related papers: Using $q$-calculus to study $LDL^T$ factorization …

The problem of polynomial regression in which the usual monomial basis is replaced by the Bernstein basis is considered. The coefficient matrix A of the overdetermined system to be solved in the least squares sense is then a rectangular…

We consider the tensor power $V=(C^N)^{\otimes n}$ of the vector representation of $gl_N$ and its weight decomposition $V=\oplus_{\lambda=(\lambda_1,...,\lambda_N)}V[\lambda]$. For $\lambda = (\lambda_1 \geq ... \geq \lambda_N)$, the…

Let $\mathbb{F}_q[T]$ be the polynomial ring over a finite field $\mathbb{F}_q$. We study the endomorphism rings of Drinfeld $\mathbb{F}_q[T]$-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings…

After introducing q-analogues of the Borel and Laplace transformations, we prove that to every formal power series solution of a linear q-difference equation with rational coefficients, we may apply several q-Borel and Laplace…

The Kalman decomposition for Linear Quantum Stochastic Systems in the real quadrature operator representation, that was derived indirectly in [1] by the authors, is derived here directly, using the "one-sided symplectic" SVD-like…

We compute the dimension $d_{n,r}(q) = \dim(\IR_q^r)$ of the defining module $\IR_q^r$ for the $q$-partition algebra. This module comes from $r$-iterations of Harish-Chandra restriction and induction on $\GL_n(\FF_q)$. This dimension is a…

We consider the problem of the R-matrix of the quantum toroidal algebra $U_{q,t}(\ddot{\mathfrak{gl}}_1)$ in the Fock representation. Using the connection between the R-matrix $R(u)$ ($u$ being the spectral parameter) and the theory of…

We investigate the homogeneous symmetric Macdonald polynomials $P_\lambda(\X;q,t)$ for the specialization $t=q^k$. We show an identity relying the polynomials $P_\lambda(\X;q,q^k)$ and $P_\lambda(\frac{1-q}{1-q^k}\X;q,q^k)$. As a…

The Tautological Lamination arises in holomorphic dynamics as a combinatorial model for the geometry of 1-dimensional slices of the Shift Locus. In each degree $q$ the tautological lamination defines an iterated sequence of partitions of…

The aim of the present study is to establish some properties for q-Bessel matrix polynomials such as several q-differential matrix equation, q-differential matrix relations and q-recurrence matrix relations, and integral representation,…

We study short sums of algebraic trace functions via the $q$-analogue of van der Corput method, and develop methods of arithmetic exponent pairs that coincide with the classical case while the moduli has sufficiently good factorizations. As…

Let $k$ be a field and let $E$ be a finite quiver. We study the structure of the finitely presented modules of finite length over the Leavitt path algebra $L_k (E)$ and show its close relationship with the finite-dimensional representations…

We show here that the refined theorems for both lecture hall partitions and anti-lecture hall compositions can be obtained as straightforward consequences of two q-Chu Vandermonde identities, once an appropriate recurrence is derived. We…

We define a reduction covariant for the representations a la Vinberg associated to stably graded Lie algebras. We then give an analogue of the LLL algorithm for the odd split special orthogonal group and show how this can be combined with…

We study the bi-commutators $[T_1, [b, T_2]]$ of pointwise multiplication and Calder\'on-Zygmund operators, and characterize their $L^{p_1}L^{p_2} \to L^{q_1}L^{q_2}$ boundedness for several off-diagonal regimes of the mixed-norm…

I continue the investigation of a q-analogue of the convolution on the line started in a joint work with Koornwinder and based on a formal definition due to Kempf and Majid. Two different ways of approximating functions by means of the…

We investigate the algebro-geometric structure of a novel two-parameter quantum deformation which exhibits the nature of a semidirect or cross-product algebra built upon GL(2) x GL(1), and is related to several other known examples of…

We study the Potts model (defined geometrically in the cluster picture) on finite two-dimensional lattices of size L x N, with boundary conditions that are free in the L-direction and periodic in the N-direction. The decomposition of the…

Inverse Vandermonde matrix calculation is a long-standing problem to solve nonsingular linear system $Vc=b$ where the rows of a square matrix $V$ are constructed by progression of the power polynomials. It has many applications in…

A finite-dimensional matrix representation of the Jackson $q$-differential operator $D_q$, defined by $D_qf(x)$ $=$ $(f(qx)-f(x))/(x(q-1))$, is written down following Calogero. Such a representation of $D_q$ should have applications in…