Note on a Paper by Robinson

The present note contains generalisations and new proofs of certain theorems in the theory of Young Tableaux and Invariant Matrices. For an account of Young Tableaux and their applications, and an introduction to the method of Clebsch-Aronhold symbols, reference should be made to Rutherford [1], and Turnbull [1], respectively. An invariant matrix T(A) of a given square matrix A is, as appears from the context in § 4 below, a matrix of polynomials in the elements of A, regarded as independent variables, such that T(AB) = T(A) T(B). Further details, and references to original sources, are given in Wallace [1].

On the astronomical method of tracking satellites

We call a contractive Hilbert space operator universal if there is a natural surjection from its generated C∗-algebra to the C∗-algebra generated by any other contraction. A universal contraction may be irreducible or a direct sum of (even nilpotent) matrices; we sharpen the latter fact in several ways, including von Neumann-type inequalities for ∗-polynomials. We also record properties of the unique C∗-algebra generated by a universal contraction and show that it can replace C∗(F2) in various Kirchberg-like reformulations of Connes' embedding problem (some known, some new). Finally we prove some analogous results for universal row contraction and universal Pythagorean C∗-algebras.

Anderson localization for radial tree graphs with random branching numbers

It is widely expected that systems which fully thermalize are chaotic in the sense of exhibiting random-matrix statistics of their energy level spacings, whereas integrable systems exhibit Poissonian statistics. In this paper, we investigate a third class: spin glasses. These systems are partially chaotic but do not achieve full thermalization due to large free energy barriers. We examine the level spacing statistics of a canonical infinite-range quantum spin glass, the quantum p-spherical model, using an analytic path integral approach. We find statistics consistent with a direct sum of independent random matrices, and show that the number of such matrices is equal to the number of distinct metastable configurations — the exponential of the spin glass “complexity” as obtained from the quantum Thouless-Anderson-Palmer equations. We also consider the statistical properties of the complexity itself and identify a set of contributions to the path integral which suggest a Poissonian distribution for the number of metastable configurations. Our results show that level spacing statistics can probe the ergodicity-breaking in quantum spin glasses and provide a way to generalize the notion of spin glass complexity beyond models with a semi-classical limit.

It is required to verify whether a given complex n × n matrix A can be made real by a similarity or a congruence transformation. Algorithms for solving these two problems are proposed and justified under the additional assumption that A is irreducible in the former case and $$A_L = \bar AA$$ is irreducible in the latter case. The irreducibility of a square complex matrix means that no unitary similarity transformation converts this matrix into a direct sum of smaller matrices. The proposed algorithms are effective in the sense that their implementation requires a finite number of arithmetic operations.

A criterion for unitary similarity of upper triangular matrices in general position

On the structure of the adjacency matrix of the line digraph of a regular digraph

1. Introduction. Let A[λ] be the irreducible invariant matrix of a general matrix of order n × n, corresponding to a partition (λ) = (λ1, λ2, …, λr) of some integer m. The problem to be discussed here is that of determining the canonical form of A[λ] when that of A is known.

The problem of estimation for a system of regression equations where the random disturbances are correlated with each other is investigated. That is, the regression equations are linked statistically, even though not structurally, through the non-diagonality of the associated variance-covariance matrix. The expression Seemingly Unrelated Regression Equations (SURE) is used to reflect the fact that the individual equations are in fact related to one another even though, superficially, they may not seem to be. The SURE model that comprises G regression equations can be written as $$ \begin{array}{*{20}{c}} {{{y}_{i}} = {{X}_{i}}{{\beta }_{i}} + {{u}_{i}},} & {i = 1, \ldots ,G,} \end{array} $$ (5.1) where the y i ∈ ℜ T are the response vectors, the \(X_i \in \Re ^{T \times k_i } \) are the exogenous matrices with full column ranks, the \(\beta _i \in \Re ^{k_i } \) are the coefficients, and the ui ∈ ℜ T are the disturbance terms. The basic assumptions underlying the SURE model (5.1) are \(E\left( {u_i } \right) = 0,\,E\left( {u_i ,u_j^T } \right) = \sigma _{i,j} I_T \) and \( {{\lim }_{{T \to \infty }}}(X_{i}^{T}{{X}_{j}}/T) \) exists (i,j = 1,…, G). In compact form the SURE model can be written as $$ \left( {\begin{array}{*{20}{c}} {{{y}_{1}}} \vdots {{{y}_{G}}} \end{array} } \right) = \left( {\begin{array}{*{20}{c}} {{{X}_{1}}} & \, & \, \, & \ddots & \, \, & \, & {{{X}_{G}}} \end{array} } \right)\left( {\begin{array}{*{20}{c}} {{{\beta }_{1}}} \vdots {{{\beta }_{G}}} \end{array} } \right) + \left( {\begin{array}{*{20}{c}} {{{u}_{1}}} \vdots {{{u}_{G}}} \end{array} } \right) $$ (5.2) where Y = (y1…y G ) and U = (u1… u G ) . The direct sum of matrices \( \oplus _{i = 1}^G X_i \) defines the GT x K block-diagonal matrix $$ \mathop{{\mathop{ \oplus }\limits_{{i = 1}} }}\limits^{G} {{X}_{i}} = {{X}_{1}} \oplus {{X}_{2}} \oplus \cdots \oplus {{X}_{G}} = \left( {\begin{array}{*{20}{c}} {{{X}_{1}}} & & & & {{{X}_{2}}} & & & & \ddots & & & & {{{X}_{G}}} \end{array} } \right), $$ (5.3) where \(K = \sum\nolimits_{i = 1}^G {k_i } \) [125]. The matrices used in the direct sum are not necessarily of the same dimension. It should be noted however, that some properties of the direct sum given in the literature are limited to square matrices [134, pages 260-261]. The set of vectors β1, β2,…, β G is denoted by {β i } The vec() operator stacks the columns of its matrix or set of vectors argument in a column vector, that is, $$ vec(Y) = \left( {\begin{array}{*{20}{c}} {y1} \vdots {{{y}_{G}}} \end{array} } \right) $$ and $$ vec({{\{ {{\beta }_{i}}\} }_{G}}) = \left( {\begin{array}{*{20}{c}} {{{\beta }_{1}}} \vdots {{{\beta }_{G}}} \end{array} } \right). $$ Hereafter, the subscript G in the set operator {•} is dropped and the direct sum of matrices \( \oplus _{i = 1}^G \) will be abbreviated to ⨁ i for notational convenience.

We completely characterize the appearance of Schur functions corresponding to partitions of the form $\nu = (1^a, b)$ (hook shapes) in the Schur function expansion of the plethysm of two Schur functions, $$s_\lambda[s_\mu] = \sum_{\nu} a_{\lambda, \mu, \nu} s_\nu.$$ Specifically, we show that no Schur functions corresponding to hook shapes occur unless $\lambda$ and $\mu$ are both hook shapes and give a new proof of a result of Carbonara, Remmel and Yang that a single hook shape occurs in the expansion of the plethysm $s_{(1^c, d)}[s_{(1^a, b)}]$. We also consider the problem of adding a row or column so that $\nu$ is of the form $(1^a,b,c)$ or $(1^a, 2^b, c)$. This proves considerably more difficult than the hook case and we discuss these difficulties while deriving explicit formulas for a special case.

A useful property of independent samples is that their correlation remains the same after applying marginal transforms. This invariance property plays a fundamental role in statistical inference, but does not hold in general for dependent samples. In this paper, we study this invariance property on the Pearson correlation coefficient and its applications. A multivariate random vector is said to have an invariant correlation if its pairwise correlation coefficients remain unchanged under any common marginal transforms. For a bivariate case, we characterize all models of such a random vector via a certain combination of comonotonicity—the strongest form of positive dependence—and independence. In particular, we show that the class of exchangeable copulas with invariant correlation is precisely described by what we call positive Fréchet copulas. In the general multivariate case, we characterize the set of all invariant correlation matrices via the clique partition polytope. We also propose a positive regression dependent model that admits any prescribed invariant correlation matrix. Finally, we show that all our characterization results of invariant correlation, except one special case, remain the same if the common marginal transforms are confined to the set of increasing ones.

We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the K K -theoretic analogue of that of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric function into Schur functions. Our expansions extend to flagged Grothendieck polynomials.

Power series expansion of a Hankel determinant

We study the explicit formula of Lusztig’s integral forms of the level one quantum affine algebraUq(sl^2)U_q(\widehat {sl}_2)in the endomorphism ring of symmetric functions in infinitely many variables tensored with the group algebra ofZ\mathbb Z. Schur functions are realized as certain orthonormal basis vectors in the vertex representation associated to the standard Heisenberg algebra. In this picture the Littlewood-Richardson rule is expressed by integral formulas, and is used to define the action of Lusztig’sZ[q,q−1]\mathbb Z[q, q^{-1}]-form ofUq(sl^2)U_q(\widehat {sl}_2)on Schur polynomials. As a result theZ[q,q−1]\mathbb Z[q, q^{-1}]-lattice of Schur functions tensored with the group algebra contains Lusztig’s integral lattice.