Complex high order Toda and Volterra lattices

Given a solution of a high order Toda lattice we construct a one parameter family of new solutions. In our method, we use a set of B¨acklund transformations in such a way that each new generalized Toda solution is related to a generalized Volterra solution.Dirección General de Investigación, Ministerio de Educación y Ciencia, MTM2006-13000-C03-02; Universidad Politécnica de Madrid; Comunidad Autónoma de Madrid CCG06-UPM/MTM- 539; CMUC/FC

NOD2 mutations and colorectal cancer - Where do we stand?

Due to the overwhelming burden of colorectal cancer (CRC), great effort has been placed on identifying genetic mutations that contribute to disease development and progression. One of the most studied polymorphisms that could potentially increase susceptibility to CRC involves the nucleotide-binding and oligomerization-domain containing 2 (NOD2) gene. There is growing evidence that the biological activity of NOD2 is far greater than previously thought and a link with intestinal microbiota and mucosal immunity is increasingly sought after. In fact, microbial composition may be an important contributor not only to inflammatory bowel diseases (IBD) but also to CRC. Recent studies have showed that deficient NOD2 function confers a communicable risk of colitis and CRC. Despite the evidence from experimental models, population-based studies that tried to link certain NOD2 polymorphisms and an increase in CRC risk have been described as conflicting. Significant geographic discrepancies in the frequency of such polymorphisms and different interpretations of the results may have limited the conclusions of those studies. Since being first associated to IBD and CRC, our understanding of the role of this gene has come a long way, and it is tempting to postulate that it may contribute to identify individuals with susceptible genetic background that may benefit from early CRC screening programs or in predicting response to current therapeutic tools. The aim of this review is to clarify the status quo of NOD2 mutations as genetic risk factors to chronic inflammation and ultimately to CRC. The use of NOD2 as a predictor of certain phenotypic characteristics of the disease will be analyzed as well

Dynamics and interpretation of some integrable systems via multiple orthogonal polynomials

High-order non symmetric difference operators with complex coefficients are considered. The correspondence between dynamics of the coefficients of the operator defined by a Lax pair and its resolvent function is established. The method of investigation is based on the analysis of the moments for the operator. The solution of a discrete dynamical system is studied. We give explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for the integrable systems

is it time to move forward?

Funding: This research was supported by Projects PEst-OE/AGR/UI0211/2011 from FCT, and COMPETE 2020, from ANI – Projetos ID&T Empresas em Copromoção, by the project “insitu.Biomas – Reinvent biomanufacturing systems by using an usability approach for in situ clinic temporary implants fabrication” with the reference POCI-01-0247-FEDER-017771, by the project “Print-on-Organs – Engineering bioinks and processes for direct printing on organs” with the reference POCI-01-0247-FEDER-033877, and by the project “Bone2Move – Development of “in vivo” experimental techniques and modelling methodologies for the evaluation of 4D scafolds for bone defect in sheep model: an integrative research approach” with the reference POCI-01-0145-FEDER-031146. Mariana Vieira Branquinho (SFRH/BD/146172/2019), Ana Catarina Sousa (SFRH/BD/146689/2019), and Rui Damásio Alvites (SFRH/BD/116118/2016), acknowledge FCT, for financial support.INTRODUCTION: Central airway obstruction (CAO) represents a pathological condition that can lead to airflow limitation of the trachea, main stem bronchi, bronchus intermedius or lobar bronchus. MAIN BODY: It is a common clinical situation consensually considered under-diagnosed. Management of patients with CAO can be difficult and deciding on the best treatment approach represents a medical challenge. This work intends to review CAO classifications, causes, treatments and its therapeutic limitations, approaching benign and malign presentations. Three illustrative cases are further presented, supporting the clinical problem under review. CONCLUSION: Management of CAO still remains a challenge. The available options are not always effective nor free from complications. A new generation of costume-tailored airway stents, associated with stem cell-based therapy, could be an option in specific clinical situations.publishersversionpublishe

The Darboux transformation and the complex Toda lattice

Abstract It is well known that each solution of the Toda lattice can be represented by a tridiagonal matrix J(t). Under certain restrictions, it is possible to obtain some new solution by using the Darboux transformation of J(t) − CI. Our goal is the extension of this fact, which is known for the real lattice, to high order complex Toda lattices as well as to the bi-infinite Toda lattice. In this latter case, we use the factorization LU for block-tridiagonal matrices. The Toda lattice We study the construction of some solutions { from another given solution {α n (t) , λ n (t)} , n ∈ Z. We consider: 1. the semi-infinite problem: S = N, λ 1 = 0, 2. the infinite problem: In [6] the semi-infinite complex problem was analyzed. In the real, infinite case, sufficient conditions for the existence of a new solution were given in The problem: obtain a similar result to the complex infinite Toda lattice. The generalized Toda lattice In a more general way, when S = N we consider the generalized Toda where we denote by J i,j (t) (respectively J p i,j (t)) the entry in the (i+1)-row and (j + 1)-column of matrix J(t) (respectively (J(t)) The generalized Toda lattice admits a Lax pair representation, i.e. a formulation in terms of the commutator of two operators, J(t) = [J(t), K(t)] = J(t)K(t) − K(t)J(t) , where we prove the existence of is another solution of (2), and Γ(t) is a solution of the Volterra lattice: 3 Relation between the generalized Toda lattice and some polynomials The matrix J(t) t defines the sequence of polynomials given by The main tools in the proof of [2, Th. 1.3]: a. We have established the dynamic behavior of P n (t, z), where C ∈ C verifies (3). The sequence Q (C) n (t, C) satisfies a threeterm recurrence relation whose coefficients define the new generalized solution J(t) = J(t, C) The new solutions and the Darboux transformation If we define and C ∈ C verifies (3), then there exist The new solution is defined by the Darboux transformation of J (1) (t) − CI, this is, The infinite Toda lattice Let us consider (1) with S = Z and take the infinite matrix The infinite Toda lattice admits also a Lax pair representation. However, in this case it is not possible to use directly the sequences of polynomials associated to J. , n ∈ N , it is possible to change the infinite recurrence relation to a semi-infinite recurrence relation, where E m , V m , m ∈ N , are 2 × 2-finite matrices. In this way, we can study the infinite case as a semi-infinite vectorial case. The vectors R n are not polynomials, but we can prove where the sequence {C n } of 2 × 2 matrices verifies and for each i = 1, 2, 3, 4, c ni is a polynomial in z , deg c ni ≤ n − 1. Taking I −1 := 1 0 0 −1 , W n := I −1 V n , n ∈ N , we can shoẇ , n = 2, 3, . . . This is, {W n , E n } is a solution of a semi-infinite matricial Toda lattice, like (1). The infinite Toda lattice and the Darboux transformation We defin

Dynamics and interpretation of some integrable systems via multiple orthogonal polynomials

High-order non-symmetric difference operators with complex coefficients are considered. The correspondence between dynamics of the coefficients of the operator defined by a Lax pair and its resolvent function is established. The method of investigation is based on the analysis of the moments for the operator. The solution of a discrete dynamical system is studied. We give explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for the integrable systems

On the full Kostant-Toda system and the discrete Korteweg-de Vries equations

The relation between the solutions of the full Kostant–Toda lattice and the discrete Korteweg–de Vries equation is analyzed. A method for constructing solutions of these systems is given. As a consequence of the matricial interpretation of this method, the transform of Darboux is extended for general Hessenberg banded matrices