Multiple orthogonality is considered in the realm of a Gauss-Borel factorization problem for a semi-infinite moment matrix. Perfect combinations of weights and a finite Borel measure are constructed in terms of M-Nikishin systems. These perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution, that can be obtained from the solution of a Gauss-Borel factorization problem for a semi-infinite matrix, which plays the role of a moment matrix. This leads to sequences of multiple orthogonal polynomials, their duals and second kind functions. It also gives the corresponding linear forms that are bi-orthogonal to the dual linear forms. Expressions for these objects in terms of determinants from the moment matrix are given, recursion relations are found, which imply a multi-diagonal Jacobi type matrix with snake shape, and results like the ABC theorem or the Christoffel-Darboux formula are re-derived in this context (using the factorization problem and the generalized Hankel symmetry of the moment matrix). The connection between this description of multiple orthogonality and the multi-component 2D Toda hierarchy, which can be also understood and studied through a Gauss-Borel factorization problem, is discussed. Deformations of the weights, natural for M-Nikishin systems, are considered and the correspondence with solutions to the integrable hierarchy, represented as a collection of Lax equations, is explored. Corresponding Lax and Zakharov-Shabat matrices as well as wave functions and their adjoints are determined. The construction of discrete flows is discussed in terms of Miwa transformations which involve Darboux transformations for the multiple orthogonality conditions. The bilinear equations are derived and the τ -function representation of the multiple orthogonality is given. further developments on the Gauss-Borel factorization and multi-component 2D Toda hierarchy see [7] and [29]. This motivated our initial research in relation with this paper; i.e., the construction of an appropriate Gauss-Borel factorization in the group of semi-infinite matrices leading to multiple orthogonality and integrability in a simultaneous manner. The main advantage of this approach lies in the application of different techniques based on the factorization problem used frequently in the theory of integrable systems. The key finding of this paper is, therefore, the characterization of a semi-infinite moment matrix whose Gauss-Borel factorization leads directly to multiple orthogonality. This makes sense when factorization can be performed, which is the case for perfect combinations (µ, w 1 , w 2 ), which allows us to consider some sets of multiple orthogonal polynomials (called ladders) very much in the same manner as in the (non multiple) orthogonal polynomial setting. The Gauss-Borel factorization of this moment matrix leads, when one takes into account the Hankel type symmetry of the moment matrix, to results like: 1. Recursion relations, 2. ABC theorems and 3. Christoffel-Darboux formulas. Th...

Given a matrix polynomial W(x), matrix bi-orthogonal polynomials with respect to the sesquilinear formwhere µ(x) is a matrix of Borel measures supported in some infinite subset of the real line, are considered. Connection formulas between the sequences of matrix bi-orthogonal polynomials with respect to •, • W and matrix polynomials orthogonal with respect to µ(x) are presented. In particular, for the case of nonsingular leading coefficients of the perturbation matrix polynomial W(x) we present a generalization of the Christoffel formula constructed in terms of the Jordan chains of W(x). For perturbations with a singular leading coefficient several examples by Durán et al are revisited. Finally, we extend these results to the non-Abelian 2D Toda lattice hierarchy. CONTENTS 1991 Mathematics Subject Classification. 42C05,15A23. Key words and phrases. Matrix orthogonal polynomials, Block Jacobi matrices, Darboux-Christoffel transformation, Block Cholesky decomposition, Block LU decomposition, quasi-determinants, non-Abelian Toda hierarchy. GA thanks financial support from the Universidad Complutense de Madrid Program "Ayudas para Becas y Contratos Complutenses Predoctorales en España 2011".MM & FM thanks financial support from the Spanish "Ministerio de Economía y Competitividad" research project MTM2012-36732-C03-01, Ortogonalidad y aproximación; teoría y aplicaciones.

In a recent phase 3 randomized trial of 700 patients with advanced urothelial cancer (JAVELIN Bladder 100; NCT02603432), avelumab/best supportive care (BSC) significantly prolonged overall survival (OS) relative to BSC alone as maintenance therapy following first-line chemotherapy. Tumor molecular profiling revealed that avelumab survival benefit was positively associated with PD-L1 expression by tumor cells, tumor mutational burden, APOBEC mutation signatures, expression of genes underlying innate and adaptive immune activity, and the number of alleles encoding high-affinity variants of activating Fc receptors. Gene expression signatures connected to tissue growth were potentially associated with reduced OS in avelumab/BSC. Individual biomarkers did not comprehensively identify patients who could benefit from therapy. These results characterize the complex biologic pathways underlying survival benefit from immune checkpoint inhibition in advanced urothelial carcinoma and suggest that multiple biomarkers may be needed to identify patients that would benefit from treatment.

The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated to an infinitedimensional group. A new set of discrete flows is considered and the corresponding Lax and Zakharov-Shabat equations are characterized. Reductions of block Toeplitz and Hankel bi-infinite matrix types are proposed and studied. Orlov-Schulman operators, string equations and additional symmetries (discrete and continuous) are considered. The continuous-discrete Lax equations are shown to be equivalent to a factorization problem as well as to a set of string equations. A congruence method to derive site independent equations is presented and used to derive equations in the discrete multicomponent KP sector (and also for its modification) of the theory as well as dispersive Whitham equations. IntroductionThis paper revisits the multicomponent 2D Toda hierarchy [30] from the point of view of the factorization problem associated to an infinite-dimensional group. Our main motivation is the recent discovery [3] of underlying integrable structures of multicomponent type in the theory of multiple orthogonal polynomials which is in turn connected to models of non-intersecting Brownian motions. Having in mind the fruitful applications of the Toda hierarchy to the theory of orthogonal polynomials and to the Hermitian random matrix model (see for instance [14]-[21]), it is expected that the formalism of multicomponent integrable hierarchies can be similarly applied to the study and characterization of multiple orthogonal polynomials and non-intersecting Brownian motions. In particular, the semiclassical (dispersionless) limit of multicomponent integrable hierarchies should be relevant for the analysis of large N ) type limits, see for instance [22]. An important piece of the technique required for these applications was recently provided by Takasaki and Takebe [27,28]. Indeed, they proved that the universal Whitham hierarchy (genus 0 case) [16] can be obtained as a particular dispersionless limit of the multicomponent KP hierarchy.The applications of the Toda hierarchy to the characterization of semiclassical limits make an essential use of the notion of string equations [14]-[21]- [10]. In recent years the formalism of string equations for dispersionless integrable hierarchies [26] has been much developed [32,19] but, to our knowledge, a similar formalism for dispersive multicomponent integrable hierarchies is not yet available. One of the main goals of this paper is to extend the formalism of string equations to multicomponent 2D Toda hierarchies. In this sense the consideration of factorization problems for these hierarchies turns to be of great help in order to introduce basic ingredients such as discrete flows, Orlov-Schulman operators and additional symmetries.The theory of the multicomponent KP hierarchy is discussed in length in the papers [15,4], see also [20] for its applications to geometric nets of conjugate type. In [30] it was noticed that τ functions of a 2N -multicomponent KP provide solutions of the N -mul...