Eugene Gutkin
We associate with a matrix over an arbitrary field an infinite family of matrices whose sizes vary from one to infinity; their entries are traces of powers of the original matrix. We explicitly evaluate the determinants of matrices in our family. The work is motivated by applications to graph spectra.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We give new definitions for the determinant over commutative ring $K$, noncommutative ring $\mathbf{K}$, noncommutative ring $\mathcal{K}$ with associative powers, over noncommutative nonassociative ring $\mathfrak{K}$, and study their properties.
In this paper we tried to condense the determinant of n square matrix to the determinant of (n - 1) square matrix with the mathematical proof.
Hankel determinants and automatic sequences are two classical subjects widely studied in Mathematics and Theoretical Computer Science. However, these two topics were considered totally independently, until in 1998, when Allouche, Peyrière, Wen and Wen proved that all the Hankel determinants of the Thue-Morse sequence are nonzero. This property allowed Bugeaud to prove that the irrationality exponents of the Thue-Morse-Mahler numbers are exactly 2. Since then, the Hankel determinants of several other automatic sequences, in particular, the paperfolding sequence, the Stern sequence, the period-d…
Proof of Convergence of a Laplace Expansion Algorithm For Calculating Recursions Satisfied by a Family of Determinants
本文证明了Laplace展开算法对任意带状Toeplitz矩阵行列式族递推关系的收敛性，解决了Evans和Hendel提出的开放问题。
The determinants of $\{\pm 1\}$-matrices are calculated by via the oriented hypergraphic Laplacian and summing over an incidence generalization of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of $0$ to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries…
A formula is presented for the determinant of the second additive compound of a square matrix in terms of coefficients of its characteristic polynomial. This formula can be used to make claims about the eigenvalues of polynomial matrices, with sign patterns as an important special case. A number of corollaries and applications of this formula are given.
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 precisely, we follow the holonomic ansatz that was proposed by Doron Zeilberger in 2007. We derive a compact and nice formula for Andrews's determinant, and use it to solve a challenge problem that we posed in a previous paper. By noting that Andrews's determinant is a special case of a two-parameter family of determinants, we find closed forms for several one-para…
We compute the determinant of $\sum_{n=1}^{N} \vec{A}^{(n)} \otimes \vec{B}^{(n)}$, where $\vec{A}^{(n)}$ is square and ${\vec{B}^{(n)}=\vec{x}^{(n)}{\vec{y}^{(n)}}^T}$ where $\vec{x}^{(n)}$ and $\vec{y}^{(n)}$ have length $N$.
In this paper, we compute asymptotics for the determinant of the combinatorial Laplacian on a sequence of $d$-dimensional orthotope square lattices as the number of vertices in each dimension grows at the same rate. It is related to the number of spanning trees by the well-known matrix tree theorem. Asymptotics for $2$ and $3$ component rooted spanning forests in these graphs are also derived. Moreover, we express the number of spanning trees in a $2$-dimensional square lattice in terms of the one in a $2$-dimensional discrete torus and also in the quartered Aztec diamond. As a consequence, we…
We study the question of finding the maximal determinant of matrices of odd order with entries {-1,1}. The most general upper bound on the maximal determinant, due to Barba, can only be achieved when the order is the sum of two consecutive squares. It is conjectured that the bound is always attained in such cases. Apart from these, only in orders 3, 7, 9, 11, 17 and 21 has the maximal value been established. In this paper we confirm the results for these orders, and add order 15 to the list. We follow previous authors in exhaustively searching for candidate Gram matrices having determinant gre…
A determinant formula of the Jones polynomial for a family of braids
本文扩展了Cohen等人的方法，将Jones多项式恢复为辫群某些子族加权邻接矩阵的行列式，并首次用平衡叠加Tait图在多项式时间内计算了无法由HOMFLYPT多项式导出的(2,q)环面结的Kauffman多项式。
Irreducibility of determinants, and Esterov's conjecture on $\mathscr{A}$-discriminants
本文研究行列式在方阵空间中的不可约性，解决Esterov关于$\mathscr{A}$-判别式的猜想，并推动对判别式组件和余维数的深入描述。
Recursions Satisfied by Families of Determinants with Applications to Resistance Distance
本文提出一种通过迭代拉普拉斯展开得到行列式族递推关系的六步半自动算法，可计算图的电阻距离闭式公式，证明了线性3-树的渐近猜想系数，同时提供相关入门综述以吸引斐波那契数研究者参与该领域。
Maximal determinants of matrices over the roots of unity
本文研究元素为ℓ次单位根的矩阵极大行列式，推广Hadamard问题，给出一般上界与构造，并重点解决ℓ=3、4情形。
$τ$-functions of certain Painlevé equations (PVI,PV,PIII) can be expressed as a Fredholm determinant. Further, the minor expansion of these determinants provide an interesting connection to Random partitions. This paper is a step towards understanding whether the $τ$-function of Painlevé II has a Fredholm determinant representation. The Ablowitz-Segur family of solutions are special one parameter solutions of Painlevé II and the corresponding $τ$-function is known to be the Fredholm determinant of the Airy Kernel. We develop a formalism for open contour in parallel to the one formulated in \ci…
Large set of orthogonal arrays (LOA) were introduced by D. R. Stinson, and it is also used to construct multimagic squares recently. In this paper, multimagic squares based on strong double LOA are further investigated. It is proved that there exists an MS$(q^{2t-1},t)$ for any prime power $q\geq 2t-1$ with $t\geq3$, which provided a new family of multimagic squares.
Pascal Determinantal Arrays and a Generalization of Rahimpour's Determinantal Identity
本文提出帕斯卡行列数组PD_k，通过行列构造推广经典帕斯卡数组，并证明了拉赫姆波尔恒等式的推广形式。
We describe an infinite family of non-Plücker cluster variables inside the double Bruhat cell cluster algebra defined by Berenstein, Fomin, and Zelevinsky. These cluster variables occur in a family of subalgebras of the double Bruhat cell cluster algebra which we call Double Rim Hook (DRH) cluster algebras. We discover that all of the cluster variables are determinants of matrices of special form. We conjecture that all the cluster variables of the double Bruhat-cell cluster algebra have similar determinant form. We notice the resemblance between our staircase diagram and Auslander-Reiten quiv…
By an old result of Cohn (1965), a Hadamard matrix of order n has no proper Hadamard submatrices of order m > n/2. We generalise this result to maximal determinant submatrices of Hadamard matrices, and show that an interval of length asymptotically equal to n/2 is excluded from the allowable orders. We make a conjecture regarding a lower bound for sums of squares of minors of maximal determinant matrices, and give evidence in support of the conjecture. We give tables of the values taken by the minors of all maximal determinant matrices of orders up to and including 21 and make some observation…
In this note we prove an assertion made by M. Levin in 1999: the Pascal matrix modulo 2 has the property that each of the square sub-matrices laying on the upper border or on the left border has determinants, computed in $\mathbb{Z}$, equal to 1 or -1.
Baur and Marsh computed the determinant of a matrix assembled from the cluster variables in a cluster algebra of type A. In this article we wish to describe two variations. On the one hand, we compute determinants of matrices assembled from the squares of the cluster variables in Baur--Marsh's matrix. One such determinant admits an interpretation as a Cayley--Menger determinant. On the other hand, we wish to present a formula for the determinant of a matrix of cluster variables in a cluster algebra of type D. This cluster algebra is associated with a marked oriented surface. As in Baur--Marsh'…
We give a combinatorial interpretation of the determinant of a matrix as a generating function over Brauer diagrams in two different but related ways. The sign of a permutation associated to its number of inversions in the Leibniz formula for the determinant is replaced by the number of crossings in the Brauer diagram. This interpretation naturally explains why the determinant of an even antisymmetric matrix is the square of a Pfaffian.
Classification of $K$-type formulas for the Heisenberg ultrahyperbolic operator $\square_s$ for $\widetilde{SL}(3,\mathbb{R})$ and tridiagonal determinants for local Heun functions
本文分类了 $\widetilde{SL}(3,\mathbb{R})$ 海森堡超双曲方程的 $K$-型公式，推广了先前结果，并证明相关三角行列式序列满足回文性质，最后从表示论角度给出 Sylvester 公式的新证明。
We show that certain integral positive definite symmetric tridiagonal matrices of determinant $n$ are in one to one correspondence with elements of $(\mathbb Z/n\mathbb Z)^*$. We study some properties of this correspondence. In a somewhat unrelated second part we discuss a construction which associates a sequence of integral polytopes to every integral symmetric matrix.
The six-vertex model with Domain Wall Boundary Conditions, or square ice, is considered for particular values of its parameters, corresponding to 1-, 2-, and 3-enumerations of Alternating Sign Matrices (ASMs). Using Hankel determinant representations for the partition function and the boundary correlator of homogeneous square ice, it is shown how the ordinary and refined enumerations can be derived in a very simple and straightforward way. The derivation is based on the standard relationship between Hankel determinants and orthogonal polynomials. For the particular sets of parameters correspon…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。