Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size. As a corollary, we obtain explicit generating sets of quantum minors for the torus-invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over ${\mathbb{Q}}$ .     

Lower Bounds for the Perron Root of a Sum of Nonnegative Matrices

Let $A^{(l)} (l = 1, \ldots ,k)$ be $n \times n$ nonnegative matrices with right and left Perron vectors $u^{(l)} $ and $v^{(l)} $ , respectively, and let $D^{(l)} $ and $E^{(l)} (l = 1, \ldots ,k)$ be positive-definite diagonal matrices of the same order. Extending known results, under the assumption that $$u^{(1)} \circ v^{(1)} = \ldots = u^{(k)} \circ v^{(k)} \ne 0$$ (where `` $ \circ $ '' denotes the componentwise, i.e., the Hadamard product of vectors) but without requiring that the matrices $A^{(l)} $ be irreducible, for the Perron root of the sum $\sum\nolimits_{l = 1}^k {D^{(l)} A^{(l)} E^{(l)} } $ we derive a lower bound of the form $$\rho \left( {\sum\limits_{l = 1}^k {D^{(l)} A^{(l)} E^{(l)} } } \right) \geqslant \sum\limits_{l = 1}^k {\beta _{l\rho } (A^{(l)} ),{\text{ }}\beta _l >0.} $$ Also we prove that, for arbitrary irreducible nonnegative matrices $A^{{\text{ (}}l{\text{)}}} (l = 1, \ldots ,k),$ , $$\rho \left( {\sum\limits_{l = 1}^k {A^{(l)} } } \right) \geqslant \sum\limits_{l = 1}^k {\alpha _{l\rho } (A^{(l)} ),} $$ where the coefficients ∝₁>0 are specified using an arbitrarily chosen normalized positive vector. The cases of equality in both estimates are analyzed, and some other related results are established. Bibliography: 8 titles.     

Hurwitz ball quotients

We consider the analogue of Hurwitz curves, smooth projective curves \(C\) of genus \(g \ge 2\) that realize equality in the Hurwitz bound \(|{{\mathrm{Aut}}}(C)| \le 84 (g - 1)\) , to smooth compact quotients \(S\) of the unit ball in \(\mathbb C^2\) . When \(S\) is arithmetic, we show that \(|{{\mathrm{Aut}}}(S)| \le 288 e(S)\) , where \(e(S)\) is the (topological) Euler characteristic, and in the case of equality show that \(S\) is a regular cover of a particular Deligne–Mostow orbifold. We conjecture that this inequality holds independent of arithmeticity, and note that work of Xiao makes progress on this conjecture and implies the best-known lower bound for the volume of a complex hyperbolic \(2\) -orbifold.     

Separation and relaxation for cones of quadratic forms

Let ${P \subseteq \mathfrak{R}_{n}}$ be a pointed, polyhedral cone. In this paper, we study the cone ${\mathcal{C} = {\rm cone}\{xx^T : x \in P\}}$ of quadratic forms. Understanding the structure of ${\mathcal{C}}$ is important for globally solving NP-hard quadratic programs over P. We establish key characteristics of ${\mathcal{C}}$ and construct a separation algorithm for ${\mathcal{C}}$ provided one can optimize with respect to a related cone over the boundary of P. This algorithm leads to a nonlinear representation of ${\mathcal{C}}$ and a class of tractable relaxations for ${\mathcal{C}}$ , each of which improves a standard polyhedral-semidefinite relaxation of ${\mathcal{C}}$ . The relaxation technique can further be applied recursively to obtain a hierarchy of relaxations, and for constant recursive depth, the hierarchy is tractable. We apply this theory to two important cases: P is the nonnegative orthant, in which case ${\mathcal{C}}$ is the cone of completely positive matrices; and P is the homogenized cone of the “box” [0, 1] ⁿ . Through various results and examples, we demonstrate the strength of the theory for these cases. For example, we achieve for the first time a separation algorithm for 5 × 5 completely positive matrices.     

The Totally Nonnegative Part of G/P is a CW Complex

The totally nonnegative part of a partial ag variety G/P has been shown in [18], [17] to be a union of semialgebraic cells. Moreover, the closure of a cell was shown in [19] to be a union of smaller cells. In this paper we provide glueing maps for each of the cells to prove that (G/P)_?0 is a CW complex. This generalizes a result of Postnikov, Speyer and the second author [15] for Grassmannians.     

Optimal relationships between L^p-norms for the Hardy operator and its dual

We obtain sharp two-sided inequalities between $L^p$ -norms $(1<p<\infty )$ of functions $\textit{Hf}$ and $H^*f$ , where $H$ is the Hardy operator, $H^*$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty ).$ In an equivalent form, it gives sharp constants in the two-sided relationships between $L^p$ -norms of functions $H\varphi -\varphi $ and $\varphi $ , where $\varphi $ is a nonnegative nonincreasing function on $(0,+\infty )$ with $\varphi (+\infty )=0.$ In particular, it provides an alternative proof of a result obtained by Kruglyak and Setterqvist (Proc Am Math Soc 136:2005–2013, 2008) for $p=2k \,\,(k\in \mathbb N )$ and by Boza and Soria (J Funct Anal 260:1020–1028, 2011) for all $p\ge 2$ , and gives a sharp version of this result for $1<p<2$ .     

Computable representations for convex hulls of low-dimensional quadratic forms

Let ${\mathcal{C}}$ be the convex hull of points ${{\{{1 \choose x}{1 \choose x}^T \,|\, x\in \mathcal{F}\subset \Re^n\}}}$ . Representing or approximating ${\mathcal{C}}$ is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. We show that if n ≤ 4 and ${\mathcal{F}}$ is a simplex, then ${\mathcal{C}}$ has a computable representation in terms of matrices X that are doubly nonnegative (positive semidefinite and componentwise nonnegative). We also prove that if n = 2 and ${\mathcal{F}}$ is a box, then ${\mathcal{C}}$ has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation-linearization technique (RLT). The simplex result generalizes known representations for the convex hull of ${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$ when ${\mathcal{F}\subset\Re^2}$ is a triangle, while the result for box constraints generalizes the well-known fact that in this case the RLT constraints generate the convex hull of ${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$ . When n = 3 and ${\mathcal{F}}$ is a box, we show that a representation for ${\mathcal{C}}$ can be obtained by utilizing the simplex result for n = 4 in conjunction with a triangulation of the 3-cube.     

Extensions of the Borel–Cantelli lemma in general measure spaces

In this paper, an important bilateral inequality for a sequence of nonnegative measurable functions on a measure space \((S,\mathcal {B}_S,\mu )\) is obtained, and some sufficient conditions for \(\mu \left( \limsup \limits _{n\rightarrow \infty }A_n\right) =\mu (S)\) are given. In addition, a weighted version of the Borel–Cantelli Lemma on the measure space is obtained. Our results generalize the corresponding ones for bounded random sequences to the case of unbounded measurable functions.     

Using quadratic convex reformulation to tighten the convex relaxation of a quadratic program with complementarity constraints

Quadratic Convex Reformulation (QCR) is a technique that has been proposed for binary and mixed integer quadratic programs. In this paper, we extend the QCR method to convex quadratic programs with linear complementarity constraints (QPCCs). Due to the complementarity relationship between the nonnegative variables $y$ and $w$ , a term $y^{T}Dw$ can be added to the QPCC objective function, where $D$ is a nonnegative diagonal matrix chosen to maintain the convexity of the objective function and the global resolution of the QPCC. Following the QCR method, the products of linear equality constraints can also be used to perturb the QPCC objective function, with the goal that the new QP relaxation provides a tighter lower bound. By solving a semidefinite program, an equivalent QPCC can be obtained whose QP relaxation is as tight as possible. In addition, we extend the QCR to a general quadratically constrained quadratic program (QCQP), of which the QPCC is a special example. Computational tests on QPCCs are presented.     

Path Methods for Strong Shift Equivalence of Positive Matrices

In the early 1990’s, Kim and Roush developed path methods for establishing strong shift equivalence (SSE) of positive matrices over a dense subring $\mathcal{U}$ of ?. This paper gives a detailed, unified and generalized presentation of these path methods. New arguments which address arbitrary dense subrings $\mathcal{U}$ of ? are used to show that for any dense subring $\mathcal{U}$ of ?, positive matrices over $\mathcal{U}$ which have just one nonzero eigenvalue and which are strong shift equivalent over $\mathcal{U}$ must be strong shift equivalent over $\mathcal{U}_{+}$ . In addition, we show matrices on a path of positive shift equivalent real matrices are SSE over ?₊; positive rational matrices which are SSE over ?₊ must be SSE over ?₊; and for any dense subring $\mathcal{U}$ of ?, within the set of positive matrices over $\mathcal{U}$ which are conjugate over $\mathcal{U}$ to a given matrix, there are only finitely many SSE- $\mathcal{U}_{+}$ classes.     

On Compact Wavelet Matrices of Rank m and of Order and Degree N

A new parametrization (one-to-one onto map) of compact wavelet matrices of rank $m$ and of order and degree $N$ is proposed in terms of coordinates in the Euclidian space $\mathbb {C}^{(m-1)N}$ . The developed method depends on Wiener–Hopf factorization of corresponding unitary matrix functions and allows to construct compact wavelet matrices efficiently. Some applications of the proposed method are discussed.     

Integer valued polynomials on lower triangular integer matrices

Let $D$ be an integral domain with quotient field $K$ . In this paper we study the algebra of polynomials in $K[x]$ which map the set of lower triangular $n\times n$ matrices with coefficients in $D$ into itself and show that it coincides with the algebra of polynomials whose divided differences of order $k$ map $D^{k+1}$ into $D$ for every $k< n$ . Using this result we describe the polynomial closure of this set of matrices when $D$ is the ring of integers in a global field.     

Cycle Lemma, Parking Functions and Related Multigraphs

For positive integers a and b, an ${(a, \overline{b})}$ -parking function of length n is a sequence (p ₁, . . . , p _n ) of nonnegative integers whose weakly increasing order q ₁ ≤ . . . ≤ q _n satisfies the condition q _i  < a + (i ? 1)b. In this paper, we give a new proof of the enumeration formula for ${(a, \overline{b})}$ -parking functions by using of the cycle lemma for words, which leads to some enumerative results for the ${(a, \overline{b})}$ -parking functions with some restrictions such as symmetric property and periodic property. Based on a bijection between ${(a, \overline{b})}$ -parking functions and rooted forests, we enumerate combinatorially the ${(a, \overline{b})}$ -parking functions with identical initial terms and symmetric ${(a, \overline{b})}$ -parking functions with respect to the middle term. Moreover, we derive the critical group of a multigraph that is closely related to ${(a, \overline{b})}$ -parking functions.     

Complexity and nonlinear semidefinite programming reformulation of \ell _1-constrained nonconvex quadratic optimization

In this article, we show that the $\ell _1$ -constrained nonconvex quadratic optimization problem (QPL1) is NP-hard, even when the off-diagonal elements are all nonnegative. Then we give an answer to Pinar and Teboulle’s open problem on the nonlinear semidefinite programming relaxation of (QPL1). The analytical approach is extended to $\ell _p$ -constrained quadratic programs with $1<p<2$ .     

Supercyclic abelian semigroups of matrices on \mathbb {C}^{n}

We give a complete characterization of a supercyclic abelian semigroup of matrices on \(\mathbb {C}^{n}\) . For finitely generated semigroups, this characterization is explicit and it is used to determine the minimal number of matrices in normal form over \(\mathbb {C}\) that form a supercyclic abelian semigroup on \({\mathbb {C}}^{n}\) . In particular, no abelian semigroup generated by \(n-1\) matrices on \(\mathbb {C}^{n}\) can be supercyclic.     

Modulus of boundary values of analytic functions of class

Let ω be a modulus of continuity, be the class of all functions analytic in the unit disk of the complex plane and such that . A condition is given (depending essentially on ω), necessary for a nonnegative function defined on the unit circle to coincide with the modulus of some function of class .     

The Root Operator on Finite Zero Based Invariant Subspaces of the Weighted Bergman Space

For a nonnegative integer α, we study and compute the root functions ${R_{\alpha}^{I}(z, w) = (1-\overline{w}z)^{2+\alpha}K_{\alpha}^{I}(z, w)}$ of finite zero based invariant subspaces I of the weighted Bergman space ${A_{\alpha}^{2}}$ , where ${K_{\alpha}^{I}}$ is the reproducing kernel of I. Furthermore, we estimate ranks of the corresponding root operators.     

Branched covers of the sphere and the prime-degree conjecture

To a branched cover ${\widetilde{\Sigma} \to \Sigma}$ between closed, connected, and orientable surfaces, one associates a branch datum, which consists of Σ and ${\widetilde{\Sigma}}$ , the total degree d, and the partitions of d given by the collections of local degrees over the branching points. This datum must satisfy the Riemann–Hurwitz formula. A candidate surface cover is an abstract branch datum, a priori not coming from a branched cover, but satisfying the Riemann– Hurwitz formula. The old Hurwitz problem asks which candidate surface covers are realizable by branched covers. It is now known that all candidate covers are realizable when Σ has positive genus, but not all are when Σ is the 2-sphere. However, a long-standing conjecture asserts that candidate covers with prime degree are realizable. To a candidate surface cover, one can associate one ${\widetilde {X} \dashrightarrow X}$ between 2-orbifolds, and in Pascali and Petronio (Trans Am Math Soc 361:5885–5920, 2009), we have completely analyzed the candidate surface covers such that either X is bad, spherical, or Euclidean, or both X and ${\widetilde{X}}$ are rigid hyperbolic orbifolds, thus also providing strong supporting evidence for the prime-degree conjecture. In this paper, using a variety of different techniques, we continue this analysis, carrying it out completely for the case where X is hyperbolic and rigid and ${\widetilde{X}}$ has a 2-dimensional Teichmüller space. We find many more realizable and non-realizable candidate covers, providing more support for the prime-degree conjecture.     

Rigidity Theorems for Manifolds with Boundary and Nonnegative Ricci Curvature

In this paper, we obtain some rigidity theorems for compact Riemannian manifolds Ω with boundary M and nonnegative Ricci curvature; for instance, we prove that the existence of certain functions on M together with a lower bound c > 0 on the principal curvtures of M imply that Ω is an euclidean ball of radius $ {1\over c} $ .