Related papers: Cauchy identities for staircase matrices
We present some identities related to the Cauchy-Schwarz inequality in complex inner product spaces. A new proof of the basic result on the subject of Strengthened Cauchy-Schwarz inequalities is derived using these identities. Also, an…
Cauchy's determinant formula (1841) involving $\det ((1-u_i v_j)^{-1})$ is a fundamental result in symmetric function theory. It has been extended in several directions, including a determinantal extension by Frobenius [J. reine angew.…
We present an analogue of the differential calculus in which the role of polynomials is played by certain ordered sets and trees. Our combinatorial calculus has all nice features of the usual calculus and has an advantage that the elements…
We use the correlation functions of vertex operators to give a proof of Cauchy's formula \begin{align*} \prod^K_{i=1}\prod^N_{j=1}(1-x_iy_j)=\sum_{\mu\subseteq [K\times N]}(-1)^{|\mu|}s_{\mu}\{x\}s_{\mu'}\{y\}. \end{align*} As an…
The partition function for unitary two matrix models is known to be a double KP tau-function, as well as providing solutions to the two dimensional Toda hierarchy. It is shown how it may also be viewed as a Borel sum regularization of…
Symmetrizable matrices are those which are symmetric when multiplied by a diagonal matrix with positive entries. The Cauchy interlace theorem states that the eigenvalues of a real symmetric matrix interlace with those of any principal…
The Cauchy-type product of two arithmetic functions $f$ and $g$ on nonnegative integers is defined as $(f\bullet g)(k):=\sum_{m=0}^{k} {k\choose m}f(m)g(k-m)$. We explore some algebraic properties of the aforementioned convolution, which is…
The Cauchy identity gives a recipe for decomposing a double Grothendieck polynomial $\mathfrak{G}^{(\beta)}_w(x;y)$ as a sum of products $\mathfrak{G}^{(\beta)}_v(x)\mathfrak{G}^{(\beta)}_u(y)$ of single Grothendieck polynomials.…
Using a specific form of the triple product identity, polygonal number identities are stated. Further number identities are examined that can be considered identities related to modular sets of numbers. The identities can be used to give…
We show that a relation among minimal non-faces of a fillable complex $K$ yields an identity of iterated (higher) Whitehead products in a polyhedral product over $K$. In particular, for the $(n-1)$-skeleton of a simplicial $n$-sphere, we…
Explicit formulas for the Gauss decomposition of elliptic Cauchy type matrices are derived in a very simple way. The elliptic Cauchy identity is an immediate corollary.
We study certain bijection between plane partitions and $\mathbb{N}$-matrices. As applications, we prove a Cauchy-type identity for generalized dual Grothendieck polynomials. We introduce two statistics on plane partitions, whose generating…
In this paper we exploit the umbral calculus framework to reformulate the so-called discrete Cauchy-Kovalevskaya extension in the scope of hypercomplex variables. The key idea is to consider not only formal power series representation for…
Motivated by statistical applications, this paper introduces Cauchy identities for characters of the compact classical groups. These identities generalize the well-known Cauchy identity for characters of the unitary group, which are Schur…
Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular…
We give non-symmetric versions of the Cauchy kernel and Littlewood's kernels, corresponding to the types $A_n$, $B_n$, $C_n$ and $D_n$, of the classical groups. We show that these new kernels are diagonal in the basis of two families of key…
We study the asymptotics of Schur polynomials with partitions $\lambda$ which are almost staircase; more precisely, partitions that differ from $((m-1)(N-1),(m-1)(N-2),\ldots,(m-1),0)$ by at most one component at the beginning as…
Given matrices $A$ and $B$ such that $B=f(A)$, where $f(z)$ is a holomorphic function, we analyze the relation between the singular values of the off-diagonal submatrices of $A$ and $B$. We provide family of bounds which depend on the…
In this series of studies on Cauchy's function $f(z)$ ($z=x+iy$) and its integral $J[f(z)]\equiv (2\pi i)^{-1}\oint_C f(t)dt/(t-z)$ taken along a Jordan contour $C$, the aim is to investigate their comprehensive properties over the entire…
We investigate the product of $n$ complex non-Hermitian, independent random matrices, each of size $N_i\times N_{i+1}$ $(i=1,...,n)$, with independent identically distributed Cauchy entries (Cauchy-Lorentz matrices). The joint probability…