相关论文: Lambda-determinants and domino-tilings
This article evaluates the determinants of two classes of special matrices, which are both from a number theory problem. Applications of the evaluated determinants can be found in [arXiv:math.NT/0509523]. Note that the two determinants are…
A square matrix is $k$-Toeplitz if its diagonals are periodic sequences of period $k$. We find universal formulas for the determinant, the characteristic polynomial, some eigenvectors, and the entries of the inverse of any tridiagonal…
We evaluate a curious determinant, first mentioned by George Andrews in 1980 in the context of descending plane partitions. Our strategy is to combine the famous Desnanot-Jacobi-Dodgson identity with automated proof techniques. More…
The Hadamard maximal determinant (maxdet) problem is to find the maximum determinant D(n) of a square {+1, -1} matrix of given order n. Such a matrix with maximum determinant is called a saturated D-optimal design. We consider some cases…
The Seidel matrix of a tournament on $n$ players is an $n\times n$ skew-symmetric matrix with entries in $\{0, 1, -1\}$ that encapsulates the outcomes of the games in the given tournament. It is known that the determinant of an $n\times n$…
The notion of the higher rank numerical range $\Lambda_{k}(L(\lambda))$ for matrix polynomials $L(\lambda)=A_{m}\lambda^{m}+...+A_{1}\lambda+A_{0}$ is introduced here and some fundamental geometrical properties are investigated. Further,…
Let $M$ be an $mn\times mn$ matrix over a commutative ring $R$. Divide $M$ into $m \times m$ blocks. Assume that the blocks commute pairwise. Consider the following two procedures: (1) Evaluate the $n \times n$ determinant formula at these…
The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P =…
The determinacy of lightface $\Delta^1_{2n+2}$ and boldface $\boldsymbol{\Pi}^1_{2n+1}$ sets implies the existence of an $(\omega, \omega_1)$-iterable $M_{2n+1}^{\#}$.
The functional determinants of the GJMS scalar operators, P_{2k}, on even-dimensional spheres are computed via Barnes multiple gamma functions relying on the numerical availability of the digamma function. For the critical k=d/2 case, it is…
Graham and Winkler derived a formula for the determinant of the distance matrix of a full-dimensional set of $n + 1$ points $\{ x_{0}, x_{1}, \ldots , x_{n} \}$ in the Hamming cube $H_{n} = ( \{ 0,1 \}^{n}, \ell_{1} )$. In this article we…
For every $2n\times 2n$ real positive definite matrix $A,$ there exists a real symplectic matrix $M$ such that $M^TAM=\diag(D,D),$ where $D$ is the $n\times n$ positive diagonal matrix with diagonal entries $d_1(A)\le \cdots\le d_n(A).$ The…
In a region $R$ consisting of unit squares, a domino is the union of two adjacent squares and a (domino) tiling is a collection of dominoes with disjoint interior whose union is the region. The flip graph $\mathcal{T}(R)$ is defined on the…
The Hamming ball of radius $w$ in $\{0,1\}^n$ is the set ${\cal B}(n,w)$ of all binary words of length $n$ and Hamming weight at most $w$. We consider injective mappings $\varphi: \{0,1\}^m \to {\cal B}(n,w)$ with the following domination…
We introduce a new notion of the determinant, called symmetrized determinant, for a square matrix with the entries in an associative algebra $\AA$. The monomial expansion of the symmetrized determinant is obtained from the standard…
Let ${\bf M}=(M_1,\ldots, M_k)$ be a tuple of real $d\times d$ matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether ${\bf M}$ possesses the following property: there exist two constants…
A set of Schwinger-Dyson equations forming constraints for at most three resolvent functions are considered for a class of Chern-Simons matter matrix models with two nodes labelled by a non-vanishing number $n$. The two cases $n=2$ and $n=…
We introduce a squarefree monomial ideal associated to the set of domino tilings of a $2\times n$ rectangle and proceed to study the associated minimal free resolution. In this paper, we use results of Dalili and Kummini to show that the…
In this article, we consider the monomial complete intersection algebra $\mathbb{K}[x,y]/\langle x^d,y^q\rangle$ in two variables. For elements $l_1,\ldots,l_{d+q-2k}$ of degree $1$, we give a formula of the deteminant of linear map from…
We prove that for any $\lambda > 1$, fixed in advance, the permanent of an $n \times n$ complex matrix, where the absolute value of each diagonal entry is at least $\lambda$ times bigger than the sum of the absolute values of all other…