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BackgroundWatermelon is an important vegetable crop with dual use of both fruit and seeds. Understanding the genetic basis of fruit quality and seed size-related traits is important for efficient marker-assisted breeding in watermelon. Linkage mapping in watermelon segregating populations using genotyping-by-sequencing (GBS) provides insights into genetic control of fruit- and seed-related traits and genome collinearity in commercial watermelon cultivars. ResultsIn the present study, we conducted QTL mapping of 12 horticulturally important traits on external and internal fruit quality and seed size/weight using segregating populations derived a cross between two commercial varieties. A high-density genetic map was developed with GBS which contained more than 6,000 SNP loci in 1,004 bins with a total map length of 1261.1 cM and average marker interval of 1.26 cM or 329 kb. Phenotypic data of fruit rind color (RC), rind stripe pattern (RSP), flesh color (FFC), fruit diameter (FD), fruit length (FL), fruit shape index (FSI), fruit weight (FW), Brix content central (BCC), Brix content edge (BCE), seed length (SL), width (SW), and weight (20SWT) were collected from two locations in two years. QTL analysis identified 47 QTL for the 12 traits, of which 24 had moderate- or major-effects, and 34 were novel QTL not identified in previous studies. The QTL for RSP were identified overlapped with previous reports, and mapped the QTL to a small interval on chromosome 6. From the detected novel QTL, we identify FD (qfd2.1), FL (qfl2.1) co-located with FSI (qfsi2.1) QTL on chromosome 2, and the minor QTL qfw3.2 co-located with previously reported fruit shape QTL (qfd3.1, qfl3.1, qfsi3.1), and SW (qsw10.1) co-located with 20SWT QTL (q20swt10.1) on chromosome 10, and 5 minor QTL (qbcc2.1, qbcc5.1, qbce2.1, qbce2.2, qbce5.1) were found to be likely new locus for Brix content.ConclusionWe conducted GBS consisting of 120 F2 individuals and developed a high-density linkage map with more than 6,000 SNP loci in 1004 bins in watermelon. We identified 47 QTL for 12 fruit and seed related traits including 34 novel QTL. Our work expands the molecular breeding toolbox for watermelon to improve the yield and fruit quality.

In this study, a new method is proposed for analysing the non-linear dynamics and stability in lane changes on highways by large buses. Unlike most of the literature associated with a simulated linear dynamic model for large buses, a verified 4DOF mechanical model with non-linear tire based on vehicle test is used in the lane-change simulation, including sinusoidal steering and steer returning. According to Jacobian matrix eigenvalues of the vehicle model, bifurcations of steady steering, sinusoidal steering and steer returning with disturbance on highways are investigated using a numerical method. The numerical simulation results reveal that Hopf bifurcations are identified in steady and sinusoidal steering conditions, which translates into an oscillatory behaviour leading to instability. Various equilibrium forms of stability are found, as well as saddle and Hopf bifurcations with disturbance ζ variety. The derived knowledge of the bifurcations set is hugely important to fully understand the actual dynamic motions of vehicles when lane changing on an even surface. It is a valuable reference for safety design of large buses to improve the traffic safety of driver–vehicle–road closed-loop system.

We present weighted Golub-Kahan-Lanczos algorithms. We demonstrate their applications to the eigenvalue problem of a product of two symmetric positive definite matrices and an eigenvalue problem for the linear response problem. A convergence analysis is provided and numerical test results are reported. As another application we make a connection between the proposed algorithms and the preconditioned conjugate gradient (PCG) method.

The sign characteristics of Hermitian matrix polynomials are discussed, and in particular an appropriate definition of the sign characteristics associated with the eigenvalue infinity. The concept of sign characteristic arises in different forms in many scientific fields, and is essential for the stability analysis in Hamiltonian systems or the perturbation behavior of eigenvalues under structured perturbations. We extend classical results by Gohberg, Lancaster, and Rodman to the case of infinite eigenvalues. We derive a systematic approach, studying how sign characteristics behave after an analytic change of variables, including the important special case of Möbius transformations, and we prove a signature constraint theorem. We also show that the sign characteristic at infinity stays invariant in a neighborhood under perturbations for even degree Hermitian matrix polynomials, while it may change for odd degree matrix polynomials. We argue that the non-uniformity can be resolved by introducing an extra zero leading matrix coefficient.

The question when a general linear time invariant control system is equivalent to a port-Hamiltonian systems is answered. Several equivalent characterizations are derived which extend the characterizations of Willems to the general non-minimal case. An explicit construction of the transformation matrices is presented. The methods are applied in the stability analysis of disc brake squeal.

We present formulas for the construction of optimal H��� controllers that can be implemented in a numerically robust way. We base the formulas on the ��-iteration developed in [6]. The controller formulas proposed here avoid the solution of algebraic Riccati equations with their problematic matrix inverses and matrix products. They are also applicable in the neighborhood of the optimal ��, where the classical formulas may call for the inverse of singular or ill-conditioned matrices. The advantages of the new formulas are demonstrated by several numerical examples.

We give several different formulations for the discrete-time linear-quadratic control problem in terms of structured eigenvalue problems, and discuss the relationships among the associated structured objects: symplectic matrices and pencils, BVD-pencils and polynomials, and the recently introduced classes of palindromic pencils and matrix polynomials. We show how these structured objects can be transformed into each other, and also how their eigenvalues, eigenvectors and invariant/deflating subspaces are related.

We study canonical forms for Hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant. In contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class. We give necessary and sufficient conditions for the existence of Hamiltonian and symplectic triangular Jordan, Kronecker and Schur forms. The presented results generalize results of Lin and Ho [17] and simplify the proofs presented there.

We present numerical methods for the solution of the optimal $H_{\infty}$ control problem. In particular, we investigate the iterative part often called the $\gamma$-iteration. We derive a method with better robustness in the presence of rounding errors than other existing methods. It remains robust in the presence of rounding errors even as $\gamma$ approaches its optimal value. For the computation of a suboptimal controller, we avoid solving algebraic Riccati equations with their problematic matrix inverses and matrix products by adapting recently suggested methods for the computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils. These methods are applicable even if the pencil has eigenvalues on the imaginary axis. We compare the new method with older methods and present several examples.

We discuss matrix pencils with a double symmetry structure that arise in the Hartree-Fock model in quantum chemistry. We derive anti-triangular condensed forms from which the eigenvalues can be read off. Ideally these would be condensed forms under unitary equivalence transformations that can be turned into stable (structure preserving) numerical methods. For the pencils under consideration this is a difficult task and not always possible. We present necessary and sufficient conditions when this is possible. If this is not possible then we show how we can include other transformations that allow to reduce the pencil to an almost anti-triangular form.

The sign characteristics of Hermitian matrix polynomials are discussed,and in particular an appropriate definition of the sign characteristics associatedwith the eigenvalue infinity. The concept of sign characteristic arisesin different forms in many scientific fields, and is essential for the stabilityanalysis in Hamiltonian systems or the perturbation behavior of eigenvaluesunder structured perturbations. We extend classical results by Gohberg,Lancaster, and Rodman to the case of infinite eigenvalues. We derive asystematic approach, studying how sign characteristics behave after ananalytic change of variables, including the important special case of Möbiustransformations, and we prove a signature constraint theorem. We alsoshow that the sign characteristic at infinity stays invariant in a neighborhoodunder perturbations for even degree Hermitian matrix polynomials,while it may change for odd degree matrix polynomials. We argue that thenon-uniformity can be resolved by introducing an extra zero leading matrixcoefficient.