Related papers: Scalar $q$-subresultants and Dickson matrices
We give a $q$-analog of middle convolution for linear $q$-difference equations with rational coefficients. In the differential case, middle convolution is defined by Katz, and he examined properties of middle convolution in detail. In this…
The method of subquotients is developed and used to determine all finite dimensional rank 2 Nichols algebras of diagonal type over an arbitrary field of characteristic zero. Key Words: Hopf algebra, Nichols algebra
We consider a class of linear ODEs of second order with variable coefficients and construct its Lie algebra of Lie group of equivalence transformations. Further we find invariants and differential invariants of this Lie algebra and by using…
We revisit a formula that connects the minimal ranks of triangular parts of a matrix and its inverse and relate the result to structured rank matrices. We also address the generic minimal rank problem.
In this note, we compute the probability that a $k\times n$ matrix can be extended to an $n\times n$ invertible matrix over $\F_q[x]$, which turns out to be $(1-q^{k-n})(1-q^{k-1-n})...(1-q^{1-n})$. Connections with Dirichlet's density…
The $q$-deformed Bannai-Ito algebra was recently constructed in the threefold tensor product of the quantum superalgebra $\mathfrak{osp}_q(1\vert 2)$. It turned out to be isomorphic to the Askey-Wilson algebra. In the present paper these…
This short note concerns with two inequalities in the geometry of gradient Ricci solitons $(g, f, \lambda )$ on a smooth manifold $M$. These inequalities provide some relationships between the curvature of the Riemannian metric $g$ and the…
We investigate second order elliptic equations \[F(\mathcal{H}u) = 0\] where the function $F\colon S(n)\to\mathbb{R}$ on the space of symmetric $n\times n$ matrices is assumed to be sublinear. There is very little to be found in the…
We present a family of analogs of the Hall-Littlewood symmetric functions in the $Q$-function algebra. The change of basis coefficients between this family and Schur's $Q$-functions are $q$-analogs of numbers of marked shifted tableaux.…
Over a composition algebra $A$, a polynomial $f(x) \in A[x]$ has a root $\alpha$ if and only $f(x)=g(x)\cdot (x-\alpha)$ for some $g(x) \in A[x]$. We examine whether this is true for general Cayley-Dickson algebras. The conclusion is that…
In [4] and [5], Folsom presents a family of modular units as higher-level analogues of the Rogers-Ramanujan $q$-continued fraction. These units are constructed from analytic solutions to the higher-order $q$-recurrence equations of Selberg.…
In a series of papers the first author and Ono connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions which are related to modular forms. Naturally it is of wide interest to find other…
We define analogues of higher derivatives for $F_q$-linear functions over the field of formal Laurent series with coefficients in $F_q$. This results in a formula for Taylor coefficients of a $F_q$-linear holomorphic function, a definition…
We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic…
Matrix-valued analogues of the little q-Jacobi polynomials are introduced and studied. For the 2x2-matrix-valued little q-Jacobi polynomials explicit expressions for the orthogonality relations, Rodrigues formula, three-term recurrence…
We prove new slope inequalities for relatively minimal fibred surfaces, showing an influence of the relative irregularity, of the unitary rank and of the Clifford index on the slope. The argument uses Xiao's method and a new Clifford-type…
The tensor rank decomposition is a useful tool for the geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able…
The type-I simple Lie-superalgebras are $sl(m|n)$ and $osp(2|2n)$. We study the quantum deformations of their untwisted affine extensions $U_q(sl(m|n)^{(1)})$ and $U_q(osp(2|2n)^{(1)})$. We identify additional relations between the simple…
Fractional calculus and q-deformed Lie algebras are closely related. Both concepts expand the scope of standard Lie algebras to describe generalized symmetries. A new class of fractional q-deformed Lie algebras is proposed, which for the…
We construct a subalgebra of the Hecke algebra of type A. This is a generalization of the group algebra of the alternating groups. All the equivalent classes of irreducible representations of the subalgebra and the q-analogue of the…