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We define cut-and-paste, a construction which, given a quadriculated disk obtains a disjoint union of quadriculated disks of smaller total area. We provide two examples of the use of this procedure as a recursive step. Tilings of a disk…
Bidiagonal matrices are widespread in numerical linear algebra, not least because of their use in the standard algorithm for computing the singular value decomposition and their appearance as LU factors of tridiagonal matrices. We show that…
Square matrices of the form $\widetilde{\mathbf{A}} =\mathbf{A} + \mathbf{e}D \mathbf{f}^*$ are considered. An explicit expression for the inverse is given, provided $\widetilde{\mathbf{A}}$ and $D$ are invertible with…
The dimension of a bipartition matrix (BPM) is the sum of the dimensions of its indecomposable factors. The dimension of an indecomposable BPM is the sum of its row, column, and entry dimensions. To compute these dimensions, we apply four…
We give an easy algorithm which calculates the generating function of the cocharacter sequence of the T-ideal of the polynomial identities of the algebra of upper triangular matrices over a field of characteristic zero. Applying this…
We state and prove an identity which connects theta series associated with binary quadratic forms of idoneal discriminants $\Delta$ and $\Delta p^2$, for $p$ a prime. Employing this identity, we extend the results of Toh by writing the…
The matrix LU factorization algorithm is a fundamental algorithm in linear algebra. We propose a generalization of the LU and LEU algorithms to accommodate the case of a commutative domain and its field of quotients. This algorithm…
A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one…
Tilings of a quadriculated annulus A are counted according to volume (in the formal variable q) and flux (in p). We consider algebraic properties of the resulting generating function Phi_A(p,q). For q = -1, the non-zero roots in p must be…
We examine the structure of the partition algebra $P_n(\delta)$ over a field $k$ of characteristic $p>0$. In particular, we describe the decomposition matrix of $P_n(\delta)$ when $n<p$ and when $n=p$ and $\delta=p-1$.
We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced…
LU-factorization of matrices is one of the fundamental algorithms of linear algebra. The widespread use of supercomputers with distributed memory requires a review of traditional algorithms, which were based on the common memory of a…
Interested in formalizing the generation of fast running code for linear algebra applications, the authors show how an index-free, calculational approach to matrix algebra can be developed by regarding matrices as morphisms of a category…
An operator C on a Hilbert space H dilates to an operator T on a Hilbert space K if there is an isometry V from H to K such that C=V^*TV. A main result of this paper is, for a positive integer d, the simultaneous dilation, up to a sharp…
Let $F$ be a real quadratic number field with discriminant $D$ and $\mathcal{O}_F$ the ring of integers in $F$. Let $\chi_F$ be the Dirichlet character associated to $F/\mathbb{Q}$. Write $L(\chi_F,s)$ for the Dirichlet L-function of…
Let $\Bigl\langle\matrix{n\cr k}\Bigr\rangle$, $\Bigl\langle\matrix{B_n\cr k}\Bigr\rangle$, and $\Bigl\langle\matrix{D_n\cr k}\Bigr\rangle$ be the Eulerian numbers in the types A, B, and D, respectively -- that is, the number of…
In this paper, operator-valued bi-free distributions are investigated. Given a subalgebra $D$ of a unital algebra $B$, it is established that a two-faced family $Z$ is bi-free from $(B, B^{\mathrm{op}})$ over $D$ if and only if certain…
Working over an algebraically closed field $k$ of any characteristic, we determine the matrix factorizations for the --- suitably graded --- triangle singularities $f=x^a+y^b+z^c$ of domestic type, that is, we assume that $(a,b,c)$ are…
This paper introduces a polynomial blind algorithm that determines when two square matrices, $A$ and $B$, are permutation similar. The shifted and translated matrices $(A+\beta I+\gamma J)$ and $(B+\beta I+\gamma J)$ are used to color the…
Given any m-dimensional dilation matrix A with rational eigenvalues, we demonstrate the existence of a digit set D such that the attractor T(A,D) of the iterated function system generated by A and D is connected. We give an easily verified…